Recent zbMATH articles in MSC 52https://www.zbmath.org/atom/cc/522022-05-16T20:40:13.078697ZWerkzeugTHE BOOK. Book review of: M. Aigner and G. M. Ziegler, Proofs from THE BOOK. 6th ed.https://www.zbmath.org/1483.000252022-05-16T20:40:13.078697Z"Ullman, Daniel H."https://www.zbmath.org/authors/?q=ai:ullman.daniel-hReview of [Zbl 1392.00001].Lozenge tilings of hexagons with holes on three crossing lineshttps://www.zbmath.org/1483.050062022-05-16T20:40:13.078697Z"Byun, Seok Hyun"https://www.zbmath.org/authors/?q=ai:byun.seok-hyunSummary: The enumeration of lozenge tilings of hexagons with holes has received much attention during the last three decades. One notable feature is that a lot of the recent development involved Kuo's graphical condensation. Motivated by \textit{M. Ciucu} et al.'s work on tilings of hexagons with a removed triad of bowties [J. Comb. Theory, Ser. A 178, Article ID 105359, 40 p. (2021; Zbl 1456.52023)], in this paper, we show that the ratio of numbers of lozenge tilings of two more general regions is expressed as a simple product formula. Our proof does not involve the graphical condensation method. The proof is short and direct. We also provide a corresponding formula for cyclically symmetric lozenge tilings. Several previous results can be easily deduced from our generalization.Catalan recursion on externally ordered bases of unit interval positroidshttps://www.zbmath.org/1483.050172022-05-16T20:40:13.078697Z"Camacho, Jan Tracy"https://www.zbmath.org/authors/?q=ai:camacho.jan-tracy"Chavez, Anastasia"https://www.zbmath.org/authors/?q=ai:chavez.anastasiaSummary: The Catalan numbers form a sequence that counts over 200 combinatorial objects. A remarkable property of the Catalan numbers, which extends to these objects, is its recursive definition; that is, we can determine the \(n\)-th object from previous ones. A matroid is a combinatorial object that generalizes the notion of linear independence with connections to many fields of mathematics. A family of matroids, called unit interval positroids (UIP), are Catalan objects induced by the antiadjacency matrices of unit interval orders. Associated to each UIP is the set of externally ordered bases, which due to \textit{M. Las Vergnas} [Eur. J. Comb. 22, No. 5, 709--721 (2001; Zbl 0984.52018)], produces a lattice after adjoining a bottom element. We study the poset of externally ordered UIP bases and the implied Catalan-induced recursion. Explicitly, we describe an algorithm for constructing the lattice of a rank-\(n\) UIP from the lattice of lower ranks. Using their inherent combinatorial structure, we define a simple formula to enumerate the bases for a given UIP.Clean tangled clutters, simplices, and projective geometrieshttps://www.zbmath.org/1483.050182022-05-16T20:40:13.078697Z"Abdi, Ahmad"https://www.zbmath.org/authors/?q=ai:abdi.ahmad"Cornuéjols, Gérard"https://www.zbmath.org/authors/?q=ai:cornuejols.gerard-p"Superdock, Matt"https://www.zbmath.org/authors/?q=ai:superdock.mattSummary: A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the \(\tau = 2\) Conjecture.
Let \(\mathcal{C}\) be a clean tangled clutter. It was recently proved that \(\mathcal{C}\) has a fractional packing of value two. Collecting the supports of all such fractional packings, we obtain what is called the core of \(\mathcal{C} \). The core is a duplication of the cuboid of a set of \(0 - 1\) points, called the setcore of \(\mathcal{C}\).
In this paper, we prove three results about the setcore. First, the convex hull of the setcore is a full-dimensional polytope containing the center point of the hypercube in its interior. Secondly, this polytope is a simplex if, and only if, the setcore is the cocycle space of a projective geometry over the two-element field. Finally, if this polytope is a simplex of dimension more than three, then \(\mathcal{C}\) has the clutter of the lines of the Fano plane as a minor.
Our results expose a fascinating interplay between the combinatorics and the geometry of clean tangled clutters.On the connectivity of spaces of three-dimensional domino tilingshttps://www.zbmath.org/1483.050192022-05-16T20:40:13.078697Z"Freire, Juliana"https://www.zbmath.org/authors/?q=ai:freire.juliana"Klivans, Caroline J."https://www.zbmath.org/authors/?q=ai:klivans.caroline-j"Milet, Pedro H."https://www.zbmath.org/authors/?q=ai:milet.pedro-h"Saldanha, Nicolau C."https://www.zbmath.org/authors/?q=ai:saldanha.nicolau-corcaoSummary: We consider domino tilings of three-dimensional cubiculated manifolds with or without boundary, including subsets of Euclidean space and three-dimensional tori. In particular, we are interested in the connected components of the space of tilings of such regions under local moves. Building on the work of the third and fourth authors we allow two possible local moves, the flip and trit. These moves are considered with respect to two topological invariants, the twist and flux.
Our main result proves that, up to refinement,
\par i) Two tilings are connected by flips and trits if and only if they have the same flux.
\par ii) Two tilings are connected by flips alone if and only if they have the same flux and twist.An asymmetric bound for sum of distance setshttps://www.zbmath.org/1483.110182022-05-16T20:40:13.078697Z"Cheong, Daewoong"https://www.zbmath.org/authors/?q=ai:cheong.daewoong"Koh, Doowon"https://www.zbmath.org/authors/?q=ai:koh.doowon"Pham, Thang"https://www.zbmath.org/authors/?q=ai:pham-van-thang.For \(E \subset \mathbb{F}_q^d\), let \(\Delta(E)\) denote the distance set determined by pairs of points in \(E\). By using additive energies of sets on a paraboloid, \textit{D. Koh} et al. [Math. Z. 297, No. 3--4, 1749--1765 (2021; Zbl 1469.52014)] proved that if \(E,F \subset \mathbb{F}_q^d\) are subsets with \(|E|\cdot |F| \gg q^{d+1/3}\), then \(|\Delta(E) + \Delta(F)| > q/2\). It was also proved that the threshold \(q^{d+1/3}\) is sharp when \(|E|=|F|\).
In this paper, the three authors provided an improvement of this result in the unbalanced case, which is essentially sharp in odd dimensions. The most important tool in their proofs is an optimal \(L^2\) restriction theorem for the sphere of zero radius.
Reviewer: Guoqing Wang (Tianjin)On the lattice Hadwiger number of superballs and some other bodieshttps://www.zbmath.org/1483.111372022-05-16T20:40:13.078697Z"Vlăduţ, Serge"https://www.zbmath.org/authors/?q=ai:vladut.sergei-gLet \(C\) be a convex body in a finite-dimensional vector space \({\mathbb R}^n\) . A Hadwiger family of \(C\) is a collection of translates of \(C\), all touching \(C\) and with pairwise disjoint interiors. The Hadwiger number (or translative kissing number) \(H(C)\) of \(C\) is the maximum number of translates in a Hadwiger family of \(C\). In the present note the author is concerned only with the lattice Hadwiger number \(H_L(C)\) defined to be the largest size of a Hadwiger family \(\{v_i + C : i \in I \}\) of \(C\) that is contained in a lattice packing \(\{v + C : v \in \Lambda \}\), where \(\Lambda\) is a full rank lattice. \textit{C. Zong} [Contemp. Math. 453, 529--548 (2008; Zbl 1144.52020)] conjectured that \(H_L(C) \ge \Omega (c^n)\) for some absolute constant \(c > 1\).
The author recently [Mosc. J. Comb. Number Theory 8, No. 2, 163--177 (2019; Zbl 1448.11124)] confirmed the conjecture for the Euclidean ball \(B^n = B_2^n\) showing that \(H_L(B^n) \ge \kappa \cdot 1.015^n\) for an absolute constant \(\kappa > 0\). In the present note he showed that the same technique can prove the conjecture for many other bodies, for example, classical superballs, i.e., unit balls \(B^n_p\) in the finite-dimensional spaces \(l_p\), \(p \ge 1\), as well as for some of their generalizations. Namely, for the superballs it holds \(H_L(B^n_p) \ge \Omega (c^n)\) with \(c = 2^M = (63 \cdot 2^{-41/7})^{1/64} > 1.0013\).
Reviewer: Kanat Abdukhalikov (Al Ain)How many zeroes? Counting solutions of systems of polynomials via toric geometry at infinityhttps://www.zbmath.org/1483.130012022-05-16T20:40:13.078697Z"Mondal, Pinaki"https://www.zbmath.org/authors/?q=ai:mondal.pinakiThe overall goal of the book under review is an exposition of the solution, by the author, of the problem of estimating the number of isolated solutions in the affine space \(K^n\) of systems of \(n\) polynomial equations in \(n\) variables with coefficients in an algebraically closed field \(K\). This estimation is obtained by an explicit characterization of systems of \(n\) polynomials supported at a given collection of \(n\) convex integral polytopes in \({\mathbb R}^n\) that have the maximum number of isolated solutions in \(K^n\). This result is a generalization of theorems of \textit{A. G. Kushnirenko} [Invent. Math. 32, 1--31 (1976; Zbl 0328.32007)], \textit{D. N. Bernstein} [Funct. Anal. Appl. 9, 183--185 (1976; Zbl 0328.32001); translation from Funkts. Anal. Prilozh. 9, No. 3, 1--4 (1975)] and \textit{A. G. Hovanskii} [Funkts. Anal. Prilozh. 12, No. 1, 51--61 (1978; Zbl 0387.14012)] who solved the corresponding problem of counting solutions in the algebraic torus \((K^*)^n\).
The exposition is self-contained and for this, the book is divided in three parts. Part 1 covers the basics, starting with the algebraic geometry requisites: affine varieties over an algebraic closed field, its regular functions, coordinate rings and morphisms between affine varieties. Next, the author introduces projective space, projective and quasiprojective varieties, rational functions and rational maps. Fiber products, complete varieties and compactifications are also treated with care, and the author illustrate several of these constructions by introducing the Segre and Veronese maps. Next come Chevalley results on the image of a morphism and constructible sets. The local theory (dimension, tangent spaces and singularities) is also recalled, with the required commutative algebra, including the completion of the local ring at a point. The introductory chapter on algebraic geometry ends with the notion of degree of a (dominant) morphism, including all the necessary field theory. The second chapter on algebraic geometry is devoted to the important notion of intersection multiplicity, where the initial focus is on the intersection multiplicity at a non-singular point of a variety. The treatment is systematic, careful, and several examples illustrate important points in the development, for instance in the case of curves. The last chapter of Part 1 is devoted to the geometry of convex polyhedra, starting from the basic definitions and properties, their associated normal fans, volumes, and Minkowski sums.
The first chapter of Part 2 introduces toric varieties, the natural ambient to formulate and prove all counting results in the remaining chapters. The topics included are just the necessary ones: Algebraic torus and their Laurent polynomials coordinate rings, toric varieties defined using polytopes, and a few properties related to toric desingularizations. This chapter ends with a nice treatment of weighted projective spaces using polytopes in \({\mathbb R}^n\). The second chapter of Part 2 gives a proof of the Bernstein-Kushnirenko-Khovanski bound on the number of isolated solutions to a generic system of \(n\) Laurent polynomials in the algebraic torus \((K^*)^n\), for \(K\) an algebraically closed field. To obtain this bound, the author recalls the notions of mixed volumes using Minkowski sums of convex polytopes, since the bound is obtained in these terms, and proves the required non-degeneracy conditions.
The two main goals of Part 3 are: First, the computation of the Milnor number of the singularity at the origin of a generic polynomial or power series, completing previous work of [Bernstein, loc. cit.], \textit{C. T. C. Wall} [J. Reine Angew. Math. 509, 1--19 (1999; Zbl 0917.14001)], [Kushnirenko, loc. cit.], [Khovanskii, loc. cit.], \textit{B. Huber} and \textit{B. Sturmfels} [Discrete Comput. Geom. 17, No. 2, 137--141 (1997; Zbl 0891.65055)] and \textit{J. M. Rojas} [J. Pure Appl. Algebra 136, No. 1, 67--100 (1999; Zbl 0943.14027)], and second, the author's extension of the theorems of Bernstein, Kushnirenko, and Khovanski to counting the number of isolated solutions of systems of \(n\) polynomial equations on \(n\) variables in the affine space \(K^n\), for \(K\) algebraically closed. The four chapters of this part give a detailed and self-contained treatment of these two objectives, with two chapters devoted to each of the questions. The last chapter of the book outlines some topics of further studies for an interested reader, ranging from toric varieties, Newton-Okounkov bodies and Newton diagrams, to variations of the Bézout problem of counting number of solutions of systems of polynomial equations or counting solutions over the field of real numbers.
Two appendices collect some results on commutative algebra that are used in the text. The book will appeal to a reader interested on the arithmetic aspects of some natural intersections and interactions between algebraic and convex geometry.
Reviewer: Felipe Zaldívar (Ciudad de México)Conic divisorial ideals and non-commutative crepant resolutions of edge rings of complete multipartite graphshttps://www.zbmath.org/1483.130272022-05-16T20:40:13.078697Z"Higashitani, Akihiro"https://www.zbmath.org/authors/?q=ai:higashitani.akihiro"Matsushita, Koji"https://www.zbmath.org/authors/?q=ai:matsushita.kojiAuthors' abstract: The first goal of the present paper is to study the class groups of the edge rings of complete multipartite graphs, denoted by \(\Bbbk[K_{r_1,\dots,r_n}]\), where \(1\leq r_1\leq \ldots\leq r_n\). More concretely, we prove that the class group of \(\Bbbk[K_{r_1,\dots,r_n}]\) is isomorphic to \(\mathbb{Z}^n\) if \(n=3\) with \(r_1\geq 2\) or \(n\geq 4\), while it turns out that the excluded cases can be deduced into Hibi rings. The second goal is to investigate the special class of divisorial ideals of \(\Bbbk[K_{r_1,\dots,r_n}]\), called conic divisorial ideals. We describe conic divisorial ideals for certain \(K_{r_1,\dots,r_n}\) including all cases where \(\Bbbk[K_{r_1,\dots,r_n}]\) is Gorenstein. Finally, we give a non-commutative crepant resolution (NCCR) of \(\Bbbk[K_{r_1,\dots,r_n}]\) in the case where it is Gorenstein.
Reviewer: Amir Mafi (Sanandaj and Tehran)Curves generating extremal rays in blowups of weighted projective planeshttps://www.zbmath.org/1483.140112022-05-16T20:40:13.078697Z"González-Anaya, Javier"https://www.zbmath.org/authors/?q=ai:gonzalez-anaya.javier"González, José Luis"https://www.zbmath.org/authors/?q=ai:gonzalez.jose-luis"Karu, Kalle"https://www.zbmath.org/authors/?q=ai:karu.kalleConsider \(X\) the blow up of a weighted projective plane at a general point (or more generally of a toric surface of Picard number one). The Mori cone of curves on \(X\) is generated by the exceptional curve and another class of non-positive self intersection, which is called a negative curve \(C\) in \(X\) when it can be chosen in fact to be the class of an irreducible curve \(C \subset X\). It is not known if every \(X\) contains a negative curve (see the Introduction of the paper under review and references therein), a question closely related with that of the property on \(X\) to be a Mori Dream Space. In this paper a unifying construction of negative curves on such \(X\)'s is provided. This construction is presented in a structured way providing, for any curve, a familiy of blowups in which it defines an extemal class in the effective cone. These blowups are classified into Mori Dream Spaces and non Mori Dream Spaces.
Reviewer: Roberto Muñoz (Madrid)Newton diagrams and the geometric degree of a polynomial mapping of \(\mathbf{C}^2\)https://www.zbmath.org/1483.140212022-05-16T20:40:13.078697Z"Masternak, Mateusz"https://www.zbmath.org/authors/?q=ai:masternak.mateuszSummary: Let \(H=(f, g) : \mathbb{C}^2\to\mathbb{C}^2\) be a polynomial mapping. We give a formula for the geometric degree of \(H\) in terms of the Newton diagrams of \(f\) and \(g\). We say that the mapping \(H\) is proper if for any compact set \(K\subset \mathbb{C}^2\) its preimage \(H^{-1}(K)\) is compact. In the paper a criterion of properness of the mapping \(H\) is also given.
For the entire collection see [Zbl 1481.26002].Gorenstein property for phylogenetic trivalent treeshttps://www.zbmath.org/1483.140882022-05-16T20:40:13.078697Z"Dinu, Rodica"https://www.zbmath.org/authors/?q=ai:dinu.rodica"Vodička, Martin"https://www.zbmath.org/authors/?q=ai:vodicka.martinThe authors investigate phylogenetic tree models: rooted trees with matrices of transitions between finitely many states assigned to edges. They consider group-based models, i.e., ones with symmetries determined by a finite abelian group action. They establish several properties of such models by investigating geometric and combinatorial objects associated to a tree model: a toric variety and a lattice polytope.
The article under review starts with the proof that for a tripod tree and a group of symmetries of even cardinality the associated algebraic variety is not projectively normal. Then the authors investigate model for \(\mathbb{Z}_3\) and claw trees, and describe facets of associated polytopes. The next result concerns the Gorenstein property for polytopes associated to a trivalent tree and a small group of symmetries: for \(\mathbb{Z}_2\times\mathbb{Z}_2\) the polytope is Gorenstein of index 4 and for \(\mathbb{Z}_3\) it is Gorenstein of index 3. Finally, the authors prove the normality of associated polytopes for claw trees and \(\mathbb{Z}_3\).
Reviewer: Maria Donten-Bury (Warszawa)Double points of free projective line arrangementshttps://www.zbmath.org/1483.140962022-05-16T20:40:13.078697Z"Abe, Takuro"https://www.zbmath.org/authors/?q=ai:abe.takuroIn this very interesting and important paper the author proves the so-called Anzis-Tohaneanu conjecture.
To be precise, let \(\mathcal{A} \subset \mathbb{P}^{2}_{\mathbb{K}}\) be an arrangement of \(d\) lines in the plane, where \(\mathbb{K}\) is an arbitrary field of characteristic zero. We denote by \(n_{2}(\mathcal{A})\) the number of double intersection points of \(\mathcal{A}\). For an intersection point \(p \in \mathcal{A}\) we define \(\mu(p):=\#\{H \in \mathcal{A} : p \in H\} - 1\).
Conjecture (Anzis-Tohaneanu). If \(\mathcal{A} \subset \mathbb{P}^{2}_{\mathbb{C}}\) is supersolvable arrangement of \(d\) lines, then \(n_{2}(\mathcal{A}) \geq d/2\).
The main result of the present paper is that the Anzis-Tohaneanu Conjecture hold for supersolvable line arrangements defined over \(\mathbb{K}\). In fact, the author provides a lower bound on the number of double points for supersolvable line arrangements.
Theorem A. Let \(\mathcal{A}\subset \mathbb{P}^{2}_{\mathbb{K}}\) be a supersolvable line arrangement having \(m+k\) lines with \(k\geq 1\). Let \(p\) be a modular point of \(\mathcal{A}\) with \(\mu(p) = m-1\). Then we have the following.
1) If \(k\leq m\), then \(n_{2}(\mathcal{A}) \geq k(m-k+1)\).
2) If \(k \geq m\), then \(n_{2}(\mathcal{A}) \geq k\).
Apart from the above theorem, which can be considered as the second main result of the paper, the author presents the following two important results. Let us define a characteristic polynomial \(\chi_{0}(A;t)\) for a projective arrangement \(\mathcal{A}\) having \(d\) lines by \[\chi_{0}(\mathcal{A};t):=t^{2}-(d-1)t + \bigg(\sum_{p \in \mathrm{Sing}(\mathcal{A})}\mu(p)\bigg)-d+1,\] where \(\mathrm{Sing}(\mathcal{A})\) denotes the set of all singular points of \(\mathcal{A}\). We say that a line arrangement \(\mathcal{A}\) is divisionally free if there is \(H \in \mathcal{A}\) such that \(\chi_{0}(A;\#\mathcal{A}^{H}-1) = 0\), where \(\mathcal{A}^{H}:= \{p \in \mathrm{Sing}(\mathcal{A}) \, : \, p \in H\}\).
Theorem B. If \(\mathcal{A}\) is a divisionally free line arrangement in the projective plane defined over an arbitrary field of characteristic zero, then \(n_{2}(\mathcal{A}) > 0\).
Finally, let us present the following technical result that is important on its own rights.
Theorem C. Let \(\mathcal{A}\) be a line arrangement in the projective plane defined over an arbitrary field of characteristic zero and \(H \in \mathcal{A}\). If \(\mathcal{A}' := \mathcal{A} \setminus \{H\}\) is free, then \(n_{2}(\mathcal{A})\geq n_{2}(H)\), where \[n_{2}(H) = \# \{ p \in \mathrm{Sing}(\mathcal{A}) \, : \, \mu(p)=1, \, p \in H\}.\] Equivalently, if you add a line to a free projective line arrangement, then that line has to contain at least one double point.
Reviewer: Piotr Pokora (Kraków)Recognizing Cartesian products of matrices and polytopeshttps://www.zbmath.org/1483.150232022-05-16T20:40:13.078697Z"Aprile, Manuel"https://www.zbmath.org/authors/?q=ai:aprile.manuel"Conforti, Michele"https://www.zbmath.org/authors/?q=ai:conforti.michele"Faenza, Yuri"https://www.zbmath.org/authors/?q=ai:faenza.yuri"Fiorini, Samuel"https://www.zbmath.org/authors/?q=ai:fiorini.samuel"Huynh, Tony"https://www.zbmath.org/authors/?q=ai:huynh.tony"Macchia, Marco"https://www.zbmath.org/authors/?q=ai:macchia.marcoSummary: The 1-\textit{product} of matrices \(S_1 \in \mathbb{R}^{m_1 \times n_1}\) and \(S_2 \in \mathbb{R}^{m_2 \times n_2}\) is the matrix in \(\mathbb{R}^{(m_1+m_2) \times (n_1n_2)}\) whose columns are the concatenation of each column of \(S_1\) with each column of \(S_2\). Our main result is a polynomial time algorithm for the following problem: given a matrix \(S\), is \(S\) a 1-product, up to permutation of rows and columns? Our main motivation is a close link between the 1-product of matrices and the Cartesian product of polytopes, which relies on the concept of slack matrix. Determining whether a given matrix is a slack matrix is an intriguing problem whose complexity is unknown, and our algorithm reduces the problem to irreducible instances. Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information in information theory. We also give a polynomial time algorithm to recognize a more complicated matrix product, called the 2-\textit{product}. Finally, as a corollary of our 1-product and 2-product recognition algorithms, we obtain a polynomial time algorithm to recognize slack matrices of 2-level matroid base polytopes.
For the entire collection see [Zbl 1465.05002].Imaginary cone and reflection subgroups of Coxeter groupshttps://www.zbmath.org/1483.200742022-05-16T20:40:13.078697Z"Dyer, Matthew J."https://www.zbmath.org/authors/?q=ai:dyer.matthew-jLet \(V\) be an \(\mathbb{R}\)-vector space equipped with a symmetric bilinear form \(\langle -,- \rangle\). Suppose \((\Phi, \Pi)\) is a based root system in \(V\) with associated Coxeter system \((W,S)\). Denote
\[
\mathscr{C} = \{v \in V \mid \langle v, \alpha \rangle \ge 0, \forall \alpha \in \Pi\}, \text{ and } \mathscr{K} = (\mathbb{R}_{\ge 0} \Pi) \cap (- \mathscr{C}).
\]
Define the imaginary cone \(\mathscr{Z}\) to be
\[
\mathscr{Z} = \bigcup_{w \in W} w \mathscr{K}.
\]
This extends the notion for Kac-Moody Lie algebras studied in [\textit{V. G. Kac}, Infinite dimensional Lie algebras. Cambridge etc.: Cambridge University Press (1990; Zbl 0716.17022)]. The paper under review provides a survey on the imaginary cone, emphasizing its relationship with reflection subgroups. There are four main results in this paper, which were unknown previously, listed as follows.
Theorem 6.3./Theorem 12.2. Let \(W^\prime\) be a reflection subgroup of \(W\), then \(\mathscr{Z}_{W^\prime} \subseteq \mathscr{Z}\), where \(\mathscr{Z}_{W^\prime}\) is the imaginary cone of \(W^\prime\).
Theorem 7.6. Suppose \(W\) is irreducible, infinite, and of finite rank. Then \(\overline{\mathscr{Z}}\) is the unique non-zero \(W\)-invariant closed pointed cone contained in \(\mathbb{R}_{\ge 0} \Pi\).
Theorem 10.3. (sketched) Suppose \(W\) is of finite rank. (a) The imaginary cone and the Tits cone is a dual pair. (b) The lattice (i.e. poset) formed by faces of \(\mathscr{Z}\) is isomorphic to the lattice formed by facial subgroups without finite components. (c) The face lattice of \(\mathscr{Z}\) is dual to that of the Tits cone. (d) If \(W^\prime\) is a facial subgroup without finite components, then its imaginary cone and its Tits cone can be described explicitly by each other.
Theorem 12.3. One has \(\mathscr{Z} = \mathbb{R}_{\ge 0} (\bigcup_{W^\prime \in \daleth} \mathscr{Z}_{W^\prime})\), where \(\daleth\) is the set of dihedral reflection subgroups of \(W\).
Besides, the hyperbolic and universal cases are discussed in Section 9. In Section 13, some motivations and applications are presented, including the dominance order, limit roots, etc. In particular, the author mentioned that a weakened version of the boundedness conjecture on Lusztig's \(a\)-function can be proved, but no more details are given. There is a list of notations at the end, which is helpful in reading.
Reviewer: Hongsheng Hu (Beijing)Anisotropic tubular neighborhoods of setshttps://www.zbmath.org/1483.280012022-05-16T20:40:13.078697Z"Chambolle, Antonin"https://www.zbmath.org/authors/?q=ai:chambolle.antonin"Lussardi, Luca"https://www.zbmath.org/authors/?q=ai:lussardi.luca"Villa, Elena"https://www.zbmath.org/authors/?q=ai:villa.elenaSummary: Let \(E \subset{{\mathbb{R}}}^N\) be a compact set and \(C\subset{{\mathbb{R}}}^N\) be a convex body with \(0\in \operatorname{int}C\). We prove that the topological boundary of the anisotropic enlargement \(E+rC\) is contained in a finite union of Lipschitz surfaces. We also investigate the regularity of the volume function \(V_E(r):=|E+rC|\) proving a formula for the right and the left derivatives at any \(r>0\) which implies that \(V_E\) is of class \(C^1\) up to a countable set completely characterized. Moreover, some properties on the second derivative of \(V_E\) are proved.Rigidity theorem for self-affine arcshttps://www.zbmath.org/1483.280102022-05-16T20:40:13.078697Z"Tetenov, A. V."https://www.zbmath.org/authors/?q=ai:tetenov.andrew-v|tetenov.andrei-viktorovich"Chelkanova, O. A."https://www.zbmath.org/authors/?q=ai:chelkanova.o-aThe authors prove that every locally self-affine arc \(\gamma\subset\mathbb{R}^2\) which admits affine shifts that are arbitrarily close to the identity is either a straight line or a segment of a parabola.
Reviewer: Peter Massopust (München)The \(L_p\) Minkowski problem for the electrostatic \(\mathfrak{p} \)-capacity for \(\mathfrak{p} \geqslant n\)https://www.zbmath.org/1483.310252022-05-16T20:40:13.078697Z"Lu, Xinbao"https://www.zbmath.org/authors/?q=ai:lu.xinbao"Xiong, Ge"https://www.zbmath.org/authors/?q=ai:xiong.geSummary: Sufficient conditions are given for the existence of solutions to the discrete \(L_p\) Minkowski problem for \(\mathfrak{p} \)-capacity when \(0<p<1\) and \(\mathfrak{p}\geqslant n\).The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacityhttps://www.zbmath.org/1483.350932022-05-16T20:40:13.078697Z"Akman, Murat"https://www.zbmath.org/authors/?q=ai:akman.murat"Gong, Jasun"https://www.zbmath.org/authors/?q=ai:gong.jasun"Hineman, Jay"https://www.zbmath.org/authors/?q=ai:hineman.jay-lawrence"Lewis, John"https://www.zbmath.org/authors/?q=ai:lewis.john-l"Vogel, Andrew"https://www.zbmath.org/authors/?q=ai:vogel.andrew-lIn this interesting paper, the authors consider two potential-theoretic problems in convex geometry.
The first main result of the paper proves a Brunn-Minkowski inequality for a nonlinear capacity, \(\text{Cap}_{\mathcal{A}}\), where \(\mathcal{A}\)-capacity is associated with a nonlinear elliptic PDE whose structure is modeled by the \(p\)-Laplace equation. More precisely, let \(1<p<n\), \(0<\lambda<1\), and \(E_1\), \(E_2\) be convex compact sets with positive \(\mathcal{A}\)-capacity. Then
\[
[\text{Cap}_{\mathcal{A}}(\lambda E_1+(1-\lambda)E_2)]^{\frac{1}{n-p}}\ge\lambda[\text{Cap}_{\mathcal{A}}(E_1)]^{\frac{1}{n-p}}+(1-\lambda)[\text{Cap}_{\mathcal{A}}(E_2)]^{\frac{1}{n-p}}.
\]
Furthermore, the authors show that if equality holds for some \(E_1\) and \(E_2\), then under certain conditions on \(\mathcal{A}\), the two sets must be homothetic.
The key ingredients of the proof include the fact that \(\{x \ : \ u(x)>t\}\) is convex for \(0<t<1\), if \(u\) is a nontrivial \(\mathcal{A}\)-harmonic capacitary function for a compact, convex set \(E\). The authors also use a maximum principle argument of [\textit{R. M. Gabriel}, J. Lond. Math. Soc. 30, 388--401 (1955; Zbl 0068.08303)], and the proof of equality is inspired by ideas in [\textit{A. Colesanti} and \textit{P. Salani}, Math. Ann. 327, No. 3, 459--479 (2003; Zbl 1052.31005); \textit{M. Longinetti}, SIAM J. Math. Anal. 19, No. 2, 377--389 (1988; Zbl 0647.31001)].
Subsequently, the authors consider a Minkowski problem for a measure associated with a compact convex set \(E\) with nonempty interior and its \(\mathcal{A}\)-harmonic capacitary function in the complement of \(E\). More precisely, denoting with \(\mu_E\) this measure, the authors consider the problem of, given a finite Borel measure \(\mu\) on \(\mathbb{S}^{n-1}\), find necessary and sufficient conditions for the existence of a set \(E\) as above, with \(\mu_E=\mu\). The authors prove that necessary and sufficient conditions for existence are the same conditions in the classical Minkowski problem for volume, and also in the work [\textit{D. Jerison}, Acta Math. 176, No. 1, 1--47 (1996; Zbl 0880.35041)], which addresses electrostatic capacity. Here, the authors are inspired by ideas from [\textit{A. Colesanti} et al., Adv. Math. 285, 1511--1588 (2015; Zbl 1327.31024); \textit{D. Jerison}, Acta Math. 176, No. 1, 1--47 (1996; Zbl 0880.35041); \textit{J. L. Lewis} and \textit{K. Nyström}, J. Eur. Math. Soc. (JEMS) 20, No. 7, 1689--1746 (2018; Zbl 1397.35088); \textit{M. Venouziou} and \textit{G. C. Verchota}, Proc. Symp. Pure Math. 79, 407--422 (2008; Zbl 1160.35022)]. Finally, using the Brunn-Minkowski inequality result from the first part of this paper, the authors prove that this problem has a unique solution, up to translation when \(p\neq n-1\), and translation and dilation when \(p=n-1\).
Reviewer: Mariana Vega Smit (Bellingham)Averages and maximal averages over Product \(j\)-varieties in finite fieldshttps://www.zbmath.org/1483.420072022-05-16T20:40:13.078697Z"Koh, Doowon"https://www.zbmath.org/authors/?q=ai:koh.doowon"Lee, Sujin"https://www.zbmath.org/authors/?q=ai:lee.sujinSummary: We study both averaging and maximal averaging problems for Product \(j\)-varieties defined by \(\prod_j = \{ x\in \mathbb{F}_q^d : \prod_{k=1}^d x_k = j \}\) for \(j \in \mathbb{F}_q^{\ast}\), where \(\mathbb{F}_q^d\) denotes a \(d\)-dimensional vector space over the finite field \(\mathbb{F}_q\) with \(q\) elements. We prove the sharp \(L^p \to L^r\) boundedness of averaging operators associated to Product \(j\)-varieties. We also obtain the optimal \(L^p\) estimate for a maximal averaging operator related to a family of Product \(j\)-varieties \(\{\prod_j\}_{j \in \mathbb{F}_q^{\ast}}\).Frames over finite fields: basic theory and equiangular lines in unitary geometryhttps://www.zbmath.org/1483.420192022-05-16T20:40:13.078697Z"Greaves, Gary R. W."https://www.zbmath.org/authors/?q=ai:greaves.gary"Iverson, Joseph W."https://www.zbmath.org/authors/?q=ai:iverson.joseph-w"Jasper, John"https://www.zbmath.org/authors/?q=ai:jasper.john"Mixon, Dustin G."https://www.zbmath.org/authors/?q=ai:mixon.dustin-gSummary: We introduce the study of frames and equiangular lines in classical geometries over finite fields. After developing the basic theory, we give several examples and demonstrate finite field analogs of equiangular tight frames (ETFs) produced by modular difference sets, and by translation and modulation operators. Using the latter, we prove that Gerzon's bound is attained in each unitary geometry of dimension \(d=2^{2l+1}\) over the field \(\mathbb{F}_{3^2}\). We also investigate interactions between complex ETFs and those in finite unitary geometries, and we show that every complex ETF implies the existence of ETFs with the same size over infinitely many finite fields.Fractional perimeters on the spherehttps://www.zbmath.org/1483.460312022-05-16T20:40:13.078697Z"Kreuml, Andreas"https://www.zbmath.org/authors/?q=ai:kreuml.andreas"Mordhorst, Olaf"https://www.zbmath.org/authors/?q=ai:mordhorst.olafSummary: This note treats several problems for the fractional perimeter or \(s\)-perimeter on the sphere. The spherical fractional isoperimetric inequality is established. It turns out that the equality cases are exactly the spherical caps. Furthermore, the convergence of fractional perimeters to the surface area as \(s \nearrow 1\) is proven. It is shown that their limit as \(s \searrow -\infty\) can be expressed in terms of the volume.Characterizations of higher order strongly generalized convex functionshttps://www.zbmath.org/1483.460752022-05-16T20:40:13.078697Z"Noor, Muhammad Aslam"https://www.zbmath.org/authors/?q=ai:noor.muhammad-aslam"Noor, Khalida Inayat"https://www.zbmath.org/authors/?q=ai:noor.khalida-inayat"Rassias, Michael Th."https://www.zbmath.org/authors/?q=ai:rassias.michael-thConsider a Hilbert space \(H\) and two transformations of the space, \(g,h:H\rightarrow H\). The authors extend the concepts of convex set and convex function by considering the behaviour of the function in the presence of the two transformations \(g\) and \(h\) of its domain. They obtain the concept of \((g,h)\)-convex subset of \(K\subseteq H\) as a set containing the point \({(1-t)h(x)}+tg(y)\), whenever \(x,y\in K\) and \(t\in [0,1]\). The properties of convexity, strong convexity, quasi-convexity, log-convexity of functions are extended in this framework of \((g,h)\)-convexity. Characterizations of each type of generalized convex functions are given. It is shown that the optimality conditions are characterized by a new class of variational inequalities in the class of generalized strongly convex functions. The operator parallelogram laws for the characterization of uniform Banach spaces are obtained as an applications of the generalized affine functions in the framework provided by the double transformation of the space using the pair \((g,h)\).
For the entire collection see [Zbl 1471.34005].
Reviewer: Gabriela Cristescu (Arad)A directional curvature formula for convex bodies in \(\mathbb{R}^n\)https://www.zbmath.org/1483.520012022-05-16T20:40:13.078697Z"F. Pereira, Fátima"https://www.zbmath.org/authors/?q=ai:pereira.fatima-fLet \(F\) be a compact convex set in \(\mathbb{R}^n\), with the origin in its interior, let \(P\) be a point on its boundary; the neighbourhood of \(P\) is given by an implicit equation.
An important question is then to calculate the curvature (and the curvature radius) of \(F\) nearby \(P\). In particular one wants to know the directional curvature, i.e. the curvature when one starts at \(P\) and walk towards a given direction.
To find a precise [and easy to use] formula for the curvature, the author considers the intersection curve between the boundary of \(F\) and a suitable plane, but the (usually) tangential plane is avoided. Technically, \(F\) is embedded into the epigraph of the functional \(f\) what enables to calculate the directional curvature by the use of \(f\) and its gradient.
Of course it is shown that the new formula is equivalent to the existing formula for implicit spatial curves and some concrete examples are presented.
Reviewer: Lienhard Wimmer (Isny)Large planar Poisson-Voronoi cells containing a given convex bodyhttps://www.zbmath.org/1483.520022022-05-16T20:40:13.078697Z"Calka, Pierre"https://www.zbmath.org/authors/?q=ai:calka.pierre"Demichel, Yann"https://www.zbmath.org/authors/?q=ai:demichel.yann"Enriquez, Nathanaël"https://www.zbmath.org/authors/?q=ai:enriquez.nathanaelSummary: Let \(K\) be a convex body in \(\mathbb{R}^2\). We consider the Voronoi tessellation generated by a homogeneous Poisson point process of intensity \(\lambda\) conditional on the existence of a cell \(K_\lambda\) which contains \(K\). When \(\lambda\rightarrow\infty\), this cell \(K_\lambda\) converges \textit{from above} to \(K\) and we provide the precise asymptotics of the expectation of its defect area, defect perimeter and number of vertices. As in Rényi and Sulanke's seminal papers on random convex hulls, the regularity of \(K\) has crucial importance and we deal with both the smooth and polygonal cases. Techniques are based on accurate estimates of the area of the Voronoi flower and of the support function of \(K_\lambda\) as well as on an Efron-type relation. Finally, we show the existence of limiting variances in the smooth case for the defect area and the number of vertices as well as analogous expectation asymptotics for the so-called Crofton cell.The \(\beta\)-Delaunay tessellation. III: Kendall's problem and limit theorems in high dimensionshttps://www.zbmath.org/1483.520032022-05-16T20:40:13.078697Z"Gusakova, Anna"https://www.zbmath.org/authors/?q=ai:gusakova.anna"Kabluchko, Zakhar"https://www.zbmath.org/authors/?q=ai:kabluchko.zakhar-a"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christophThis paper is the third one in a series about the \(\beta\)-Delaunay tessellation (numbers 1,2,4 are available at \url{arXiv:2005.13875v2}, \url{arXiv:2101.11316}, \url{arXiv.2108.09472}). The construction of such a tessellation is based on a Poisson point process in the space \({\mathbb R}^{d-1}\times [0,\infty)\). Its intensity measure \(\mu_\beta\) has, with respect to Lebesgue measure, a density of the form \((v,h)\mapsto \gamma c_{d,\beta}\cdot h^\beta\), where \(\beta>-1\), \(\gamma>0\), and \(c_{d,\beta}\) is a normalizing constant. The `paraboloid hull process' is then used to derive a stationary random tessellation \({\mathcal D}_\beta\) of \({\mathbb R}^{d-1}\) into simplices (alternatively, Laguerre tessellations can be employed to construct \({\mathcal D}_\beta\)). The classical Poisson--Delaunay tessellation can be considered as the limit case when \(\beta\to -1\). The \(\nu\)-weighted typical cell \(Z_{\beta,\nu}\) of \({\mathcal D}_\beta\), weighted by the \(\nu\)th power of the volume (for some \(\nu\ge -1\)), is introduced by using Palm calculus for marked point processes.
The first part of this paper treats Kendall's problem for the \(\nu\)-weighted typical cell: under the condition that \(Z_{\beta,\nu}\) has large volume, an estimate is given showing that the probability that \(Z_{\beta,\nu}\) has large deviation from a regular simplex must be small. Also in this part, it is shown that \({\mathbb P}(\mathrm{Vol}(Z_{\beta,\nu})\le a)\) behaves like a power of \(a\) as \(a\to 0\), and that it decays exponentially as \(a\to\infty\).
The last part of the paper deals with the asymptotic behavior of \(\log\mathrm{Vol}(Z_{\beta,\nu})\) as \(d+2\beta+\nu\to\infty\) (with a particular view to case where \(\beta,\nu\) are constant and the dimension tends to infinity). The cumulant method and mod-\(\phi\) convergence are used. Obtained are a central limit theorem with Berry-Esseen bound, a moderate deviation principle, concentration inequalities, and large deviation behavior.
Reviewer: Rolf Schneider (Freiburg im Breisgau)The volume of simplices in high-dimensional Poisson-Delaunay tessellationshttps://www.zbmath.org/1483.520042022-05-16T20:40:13.078697Z"Gusakova, Anna"https://www.zbmath.org/authors/?q=ai:gusakova.anna"Thäle, Christoph"https://www.zbmath.org/authors/?q=ai:thale.christophSummary: Typical weighted random simplices \(Z_\mu\), \(\mu \in (-2, \infty)\), in a Poisson-Delaunay tessellation in \(\mathbb{R}^n\) are considered, where the weight is given by the \((\mu +1)\)st power of the volume. As special cases this includes the typical (\(\mu =-1\)) and the usual volume-weighted (\(\mu =0\)) Poisson-Delaunay simplex. By proving sharp bounds on cumulants it is shown that the logarithmic volume of \(Z_\mu\) satisfies a central limit theorem in high dimensions, that is, as \(n\rightarrow \infty\). In addition, rates of convergence are provided. In parallel, concentration inequalities as well as moderate deviations are studied. The set-up allows the weight \(\mu =\mu(n)\) to depend on the dimension \(n\) as well. A number of special cases are discussed separately. For fixed \(\mu\) also mod-\(\phi\) convergence and the large deviations behaviour of the logarithmic volume of \(Z_\mu\) are investigated.The Orlicz-Minkowski problem for polytopeshttps://www.zbmath.org/1483.520052022-05-16T20:40:13.078697Z"Jiang, Meiyue"https://www.zbmath.org/authors/?q=ai:jiang.meiyue"Wang, Chu"https://www.zbmath.org/authors/?q=ai:wang.chuThe classical Minkowski problem and various modern versions of it, known as Minkowski-type problems, are crucial in the study of convex bodies throughout the 20th and 21st century and even more so in the last few decades. Some of the most influential Minkowski-type problems include the \(L_p\) Minkowski problem (which contains the well-known logarithmic Minkowski problem) and the recently posed dual Minkowski problems. These problems, with enough regularity assumed, reduce to the study of a family of Monge-Ampere type equations. The uniqueness of these solutions are often tied to isoperimetric inequalities and spetrum analysis of certain second order nonlinear elliptic operators.
The current paper establishes some existence result for the Orlicz Minkowski problem. It uses a variational approach and studies particularly the discrete case.
Reviewer: Yiming Zhao (Syracuse)The \(\phi \)-Brunn-Minkowski inequalities for general convex bodieshttps://www.zbmath.org/1483.520062022-05-16T20:40:13.078697Z"Lai, DanDan"https://www.zbmath.org/authors/?q=ai:lai.dandan"Jin, HaiLin"https://www.zbmath.org/authors/?q=ai:jin.hailinThe \(L_p\) (where \(p\geq1\)) Brunn-Minkowski theory was introduced by Lutwak which has received widespread attention. Since the papers [\textit{K. Böröczky jun.} et al., Adv. Math. 231, No. 3--4, 1974--1997 (2012; Zbl 1258.52005); J. Am. Math. Soc. 26, No. 3, 831--852 (2013; Zbl 1272.52012)], many works focus on the corresponding \(L_p\)-Brunn-Minkowski theory when \(p<1\). However, there are many open problems in this hot topic.
Inspired by the work of \textit{L. Ma} [Geom. Dedicata 177, 75--82 (2015; Zbl 1396.52014)], the authors of the present paper give a new approach to the log-Minkowski inequality (i.e., \(p=0\)) when the planar convex bodies are in dilation position. Moreover, the authors also extend \(L_p\)-Brunn-Minkowski inequality and \(L_p\)-Minkowski inequality of origin-symmetric planar convex bodies for \(0<p<1\) to their corresponding Orlicz cases.
Reviewer: Niufa Fang (Chongqing)An epigraphical approach to the representer theoremhttps://www.zbmath.org/1483.520072022-05-16T20:40:13.078697Z"Duval, Vincent"https://www.zbmath.org/authors/?q=ai:duval.vincentA new principle meant to describe solutions to convex variational problems involving finitely many measurements is proposed by means of an epigraphical approach to the celebrated representer theorem. The optimality of the new method is analyzed on several problems concerning the recovery of Radon measures. An appendix with several additional results closes the paper.
Reviewer: Sorin-Mihai Grad (Paris)Some generalizations of the shadow problem in the Lobachevsky spacehttps://www.zbmath.org/1483.520082022-05-16T20:40:13.078697Z"Kostin, A. V."https://www.zbmath.org/authors/?q=ai:kostin.andrey-viktorovichA set \(U\) in the \(n\)-dimensional Euclidean space \(E^n\) is called \(m\)-convex with respect to a point \(P \notin E^n \setminus U\) for some \(1 \leq m < n\) if there is no \(m\)-dimensional subspace containing \(P\) and disjoint from \(U\), and \(U\) is called \(m\)-semiconvex with respect to \(P\) if there is no \(m\)-dimensional half subspace containing \(P\) and disjoint from \(U\). The shadow problem asks for the minimum number \(N\) of open/closed balls mutually disjoint and disjoint from a point \(P\) such that their union is \(1\)-convex with respect to \(P\) and their centers lie on the same sphere centered at \(P\).
The aim of the author is to present solutions for variants of this problem in the hyperbolic space \(H^n\). After defining \(m\)-convexity and \(m\)-semiconvexity analogously to the Euclidean definition, he examines variants of the shadow problem in the \(2\)- and \(3\)-dimensional hyperbolic space. More specifically, he proves that the minimum number of open (closed) disjoint disks in the hyperbolic plane whose centers are at the same distance from some point \(P\), are disjoint from \(P\) and their union is \(1\)-convex with respect to \(P\), is \(2\), and the same holds if we replace disks by horodisks. Furthermore, for both open (closed) disks or horodisks this number is \(3\) if we replace \(1\)-convexity by \(1\)-semiconvexity.
For points in the \(3\)-dimensional hyperbolic space the author remarks that analogously to the Euclidean problem it can be shown that the minimum number of pairwise disjoint open (closed) balls in \(H^3\) whose centers are on a sphere centered at a point \(P\), and are disjoint from \(P\) and their union is \(1\)-convex with respect to \(P\), is \(4\). Furthermore, he proves that if we drop the condition that the centers of the balls lie on a sphere centered at \(P\), or assume that they lie on an ellipsoid centered at \(P\), this number is \(3\). He also proves results about the case when some or all of the balls are replaced by horoballs.
Reviewer: Zsolt Lángi (Budapest)Internal and external duality in abstract polytopeshttps://www.zbmath.org/1483.520092022-05-16T20:40:13.078697Z"Cunningham, Gabe"https://www.zbmath.org/authors/?q=ai:cunningham.gabe"Mixer, Mark"https://www.zbmath.org/authors/?q=ai:mixer.markSummary: We define an abstract regular polytope to be internally self-dual if its self-duality can be realized as one of its symmetries. This property has many interesting implications on the structure of the polytope, which we present here. Then, we construct many examples of internally self-dual polytopes. In particular, we show that there are internally self-dual regular polyhedra of each type \(\{p, p\}\) for \(p \geq 3\) and that there are both infinitely many internally self-dual and infinitely many externally self-dual polyhedra of type \(\{p, p\}\) for \(p\) even. We also show that there are internally self-dual polytopes in each rank, including a new family of polytopes that we construct here.Tight chiral polytopeshttps://www.zbmath.org/1483.520102022-05-16T20:40:13.078697Z"Cunningham, Gabe"https://www.zbmath.org/authors/?q=ai:cunningham.gabe"Pellicer, Daniel"https://www.zbmath.org/authors/?q=ai:pellicer.danielA chiral polytope with Schläfli symbol \(\{p_1,\, \dots,\, p_{n-1}\}\) is tight whenever it has exactly \(2p_1\cdots p_{n- 1}\) flags. In papers [Electron. J. Comb. 24, No. 3, Research Paper P3.59, 14 p. (2017; Zbl 1372.52013); Combinatorica 38, No. 1, 115--142 (2018; Zbl 1413.52016)], the first author proved the non-existence of tight chiral \(n\)-polytopes for \(n\ge 6\), determined Schläfli symbols of tight chiral polyhedra, and classified the so-called atomic chiral polyhedra which could be covered by each tight chiral polyhedron.
The present paper proves the non-existence of tight chiral 5-polytopes, defines atomic chiral polytopes which generalize atomic chiral polyhedra, and gives all atomic chiral 4-polytopes. In the same way, as every tight chiral polytope covers an atomic chiral polytope, all tight chiral 4-polytopes are classified to a great extent.
By a series of these papers, the classification of tight chiral polytopes for all ranks is roughly done, and it may provide a clue for classifying abstract polytopes with other specific properties.
Reviewer: Wei-Juan Zhang (Xi'an)A generalization of a theorem of Whitehttps://www.zbmath.org/1483.520112022-05-16T20:40:13.078697Z"Batyrev, Victor"https://www.zbmath.org/authors/?q=ai:batyrev.victor-v"Hofscheier, Johannes"https://www.zbmath.org/authors/?q=ai:hofscheier.johannesSummary: An \(m\)-dimensional simplex \(\Delta\) in \(\mathbb{R}^m\) is called \emph{empty lattice simplex} if \(\Delta\cap\mathbb{Z}^m\) is exactly the set of vertices of \(\Delta \). A theorem of White states that if \(m=3\) then, up to an affine unimodular transformation of the lattice \(\mathbb{Z}^m\), any empty lattice simplex \(\Delta\subset\mathbb{R}^3\) is isomorphic to a tetrahedron whose vertices have third coordinate 0 or 1. We prove a generalization of this theorem for some special empty lattice simplices of arbitrary odd dimension \(m=2d-1\) which was conjectured by Sebő and Borisov. Our result implies a classification of all \(2d\)-dimensional isolated Gorenstein cyclic quotient singularities with minimal \(\log \)-discrepancy \(\ge d\).\(\mathrm{SL}(n)\) covariant vector-valued valuations on \(L^p\)-spaceshttps://www.zbmath.org/1483.520122022-05-16T20:40:13.078697Z"Wang, Wei"https://www.zbmath.org/authors/?q=ai:wang.wei.16"He, Rigao"https://www.zbmath.org/authors/?q=ai:he.rigao"Liu, Lijuan"https://www.zbmath.org/authors/?q=ai:liu.lijuanThe main result of the article under review shows that the moment vector is essentially the only continuous \(\mathrm{SL}(n)\) covariant vector-valued valuation on \(L^p(\mathbb{R}^n,|x|dx)\). In more detail, the main result reads as follows:
Let \(n \geq 2\). A function \(\Phi : L^p (\mathbb{R}^n, |x|dx)\to\mathbb{R}^n\) is a continuous \(\mathrm{SL}(n)\) covariant valuation if and only if there exists a continuous function \(H:\mathbb{R}\to\mathbb{R}\) satisfying the property \(|H(\alpha)|\leq \gamma |\alpha|^p\) for all \(\alpha\in\mathbb{R}\) and some constant \(\gamma\geq 0\) such that \(\Phi(f) = m(H\circ f)\) for every \(f \in L^p (\mathbb{R}^n, |x|dx)\).
Here \(m(g)\) stands for the moment vector of \(g\in L^1 (\mathbb{R}^n, |x|dx)\) and is defined by \(m(g) = \int_{\mathbb{R}^n}g(x)xdx\), and a function \(\Phi: L^p(\mathbb{R}^ n, |x|dx)\to \mathbb{R}^n\) is called
\begin{itemize}
\item a valuation if \(\Phi(f\vee g) + \Phi(f\wedge g) = \Phi(f) + \Phi(g)\) for all \(f,g\in L^p(\mathbb{R}^n,|x|dx)\), where \(f\vee g = \max\{ f,g\}\), \(f\wedge g = \min\{ f,g\}\);
\item continuous if \(\Phi(f_i)\to \Phi(f)\) in \(\mathbb{R}^n\), as \(f_i\to f\) in \(L^p(\mathbb{R}^n, |x|dx)\);
\item \(\mathrm{SL}(n)\) covariant if \(\Phi(f\circ \phi^{-1}) = \phi\Phi(f)\) for every \(f\in L^p(\mathbb{R}^n, |x|dx)\) and every \(\phi\in \mathrm{SL}(n)\), where \(\phi^{-1}\) denotes the inverse of \(\phi\).
\end{itemize}
Reviewer: Victor Alexandrov (Novosibirsk)Incidences between points and curves with almost two degrees of freedomhttps://www.zbmath.org/1483.520132022-05-16T20:40:13.078697Z"Sharir, Micha"https://www.zbmath.org/authors/?q=ai:sharir.micha"Solomon, Noam"https://www.zbmath.org/authors/?q=ai:solomon.noam"Zlydenko, Oleg"https://www.zbmath.org/authors/?q=ai:zlydenko.olegSummary: We study incidences between points and (constant-degree algebraic) curves in three dimensions, taken from a family \(\mathcal{C}\) of curves that have \textit{almost two degrees of freedom}, meaning that (i) every pair of curves of \(\mathcal{C}\) intersect in \(O(1)\) points, (ii) for any pair of points \(p, q\), there are only \(O(1)\) curves of \(\mathcal{C}\) that pass through both points, and (iii) there exists a 6-variate real polynomial \(F\) of constant degree, so that a pair \(p, q\) of points admit a curve of \(\mathcal{C}\) that passes through both of them if and only if \(F(p, q) = 0\). (As an example, the family of unit circles in \(\mathbb{R}^3\) that pass through some fixed point is such a family.)
We begin by studying two specific instances of this scenario. The first instance deals with the case of unit circles in \(\mathbb{R}^3\) that pass through some fixed point (so called \textit{anchored} unit circles). In the second case we consider tangencies between \textit{directed points} and circles in the plane, where a directed point is a pair \((p, u)\), where \(p\) is a point in the plane and \(u\) is a direction, and \((p, u)\) is tangent to a circle \(\gamma\) if \(p \in \gamma\) and \(u\) is the direction of the tangent to \(\gamma\) at \(p\). A lifting transformation due to \textit{J. S. Ellenberg} et al. [Discrete Anal. 2016, Paper No. 18, 22 p. (2016; Zbl 1405.52025)] maps these tangencies to incidences between points and curves (`lifted circles') in three dimensions. In both instances we have a family of curves in \(\mathbb{R}^3\) with almost two degrees of freedom.
We show that the number of incidences between \(m\) points and \(n\) anchored unit circles in \(\mathbb{R}^3\), as well as the number of tangencies between \(m\) directed points and \(n\) arbitrary circles in the plane, is \(O( m^{3 / 5} n^{3 / 5} + m + n)\).
We then derive a similar incidence bound, with a few additional terms, for more general families \(\mathcal{C}\) of curves in \(\mathbb{R}^3\) with almost two degrees of freedom, under a few additional natural assumptions.
The proofs follow standard techniques, based on polynomial partitioning, but they face a critical novel issue involving the analysis of surfaces that are infinitely ruled by the respective family of curves, as well as of surfaces in a dual three-dimensional space that are infinitely ruled by the respective family of suitably defined dual curves. We either show that no such surfaces exist, or develop and adapt techniques for handling incidences on such surfaces.
The general bound that we obtain is
\[
I(P, C) = O \left( m^{3 / 5} n^{3 / 5} + ( m^{11 / 15} n^{2 / 5} + n^{8 / 9} ) \delta^{1 / 3} + m^{2 / 3} n^{1 / 3} \pi^{1 / 3} + m + n \right),
\]
where \(\pi \) (resp., \( \delta )\) is the maximal number of curves (resp., dual curves) that lie on a common surface that is infinitely ruled by the family of these curves.Tverberg's theorem, disks, and Hamiltonian cycleshttps://www.zbmath.org/1483.520142022-05-16T20:40:13.078697Z"Soberón, Pablo"https://www.zbmath.org/authors/?q=ai:soberon.pablo"Tang, Yaqian"https://www.zbmath.org/authors/?q=ai:tang.yaqian\textit{H. Tverberg} proved [J. Lond. Math. Soc. 41, 123--128 (1966; Zbl 0131.20002)] that for any set of \((r-1)(d+1)+1\) points in \(\mathbb{R}^d\) there exists a partition of them into \(r\) parts whose convex hulls intersect Such partitions are called Tverberg partitions. The paper studies a variant of Tverberg partitions. For a segment \(e\) in \(\mathbb{R}^d\) with endpoints \(x, y\), we denote by \(D(e)\) the closed ball for which \(e\) is a diameter. Given a finite set of points in \(\mathbb{R}^d\), instead of looking at the convex hulls of its subsets, the paper is interested in the balls spanned by pairs of the points. Given a graph on a finite point set as a vertex set, the graph is called Tverberg, if the intersection of balls associated with the edges of the graph is non-empty. The main result of the paper is that a finite point set of odd cardinality has a Tverberg Hamiltonian cycle, while a finite point set of even cardinality has a Tverberg Hamiltonian path.
Reviewer: László A. Székely (Columbia)Maximal density of sphere packings in dimensions 8 and 24 [after Maryna S. Viazovska et al.]https://www.zbmath.org/1483.520152022-05-16T20:40:13.078697Z"Oesterlé, Joseph"https://www.zbmath.org/authors/?q=ai:oesterle.josephSummary: The maximal density of sphere packings in a euclidean space was until recently known only in dimensions 1, 2 and 3. A young Ukrainian mathematician, \textit{M. S. Viazovska}, determined it in [Ann. Math. (2) 185, No. 3, 991--1015 (2017; Zbl 1373.52025)] in dimension 8 and soon later, in collaboration with other mathematicians, in dimension 24 [\textit{H. Cohn} et al., Ann. Math. (2) 185, No. 3, 1017--1033 (2017; Zbl 1370.52037)]. This maximal density is achieved in dimension 8 when the centers of the spheres form a root lattice of type \(E_8\), in dimension 24 when they form a Leech lattice. In both cases, these are the only periodical sphere packings with maximal density (up to homothety and isometry).
For the entire collection see [Zbl 1416.00029].Nested triacontahedral shells or how to grow a quasi-crystalhttps://www.zbmath.org/1483.520162022-05-16T20:40:13.078697Z"Longuet-Higgins, Michael S."https://www.zbmath.org/authors/?q=ai:longuet-higgins.michael-s(no abstract)On connectedness of discretized setshttps://www.zbmath.org/1483.520172022-05-16T20:40:13.078697Z"Brimkov, Boris"https://www.zbmath.org/authors/?q=ai:brimkov.boris"Brimkov, Valentin E."https://www.zbmath.org/authors/?q=ai:brimkov.valentin-eIn many areas of mathematics and its applications, a discretization of a set \(X\subseteq \mathbb{R}^n\) is desirable. A frequently used, natural and simple type of discretization of a set \(X\) is the one defined by the integer points within a closed neighborhood of \(X\) of radius \(r\). This radius \(r\) is called the offset radius. The paper under review determines the minimum threshold for the offset radius, beyond which the discretization of an arbitrary (possibly disconnected) set is always connected.
For the entire collection see [Zbl 1464.68015].
Reviewer: László A. Székely (Columbia)A Pu-Bonnesen inequalityhttps://www.zbmath.org/1483.530852022-05-16T20:40:13.078697Z"Katz, Mikhail G."https://www.zbmath.org/authors/?q=ai:katz.mikhail-g"Sabourau, Stéphane"https://www.zbmath.org/authors/?q=ai:sabourau.stephaneThe classical Pu systolic inequality establishes that
\[
\mathrm{area}(g)-\frac{2}{\pi}\mathrm{sys}(g)^2\geq 0
\]
for every Riemannian metric \(g\) on the real projective plane \({\mathbb R}{\mathbb P}^2\). In the paper under review the authors prove a Bonnesen-type inequality that generalizes Pu's inequality, in terms of the circumscribed \(R\) and inscribed \(r\) radii of the Euclidean embedding of the orientable double cover \(S_g\subset{\mathbb R}^3\). More precisely, they show that if \(({\mathbb R}{\mathbb P}^2,g)\) has positive Gaussian curvature, then there exists a monotone continuous function \(\lambda(t)>0\) for \(t>0\) such that
\[
\dfrac{\mathrm{area}(g)}{\mathrm{sys}(g)^2}-\dfrac{2}{\pi}\geq\lambda\left(\dfrac{R-r}{\mathrm{ sys}}\right),
\]
where \(\lambda(t)\) is asymptotically linear as \(t\to\infty\).
The authors present two different proofs of this result, using an extrinsic argument and an intrinsic approach. The first one exploits powerful tools from convexity such as John's ellipsoid theorem or Blaschke's selection theorem.
Reviewer: Maria A. Hernández Cifre (Murcia)Convexity in topological betweenness structureshttps://www.zbmath.org/1483.540172022-05-16T20:40:13.078697Z"Anderson, Daron"https://www.zbmath.org/authors/?q=ai:anderson.daron"Bankston, Paul"https://www.zbmath.org/authors/?q=ai:bankston.paul"McCluskey, Aisling"https://www.zbmath.org/authors/?q=ai:mccluskey.aisling-eA betweenness structure is a pair \(\langle X,[\cdot,\cdot,\cdot] \rangle\), where \(X\) is a set and \([\cdot,\cdot,\cdot]\subset X^{3}\) is a ternary relation satisfying that
\begin{itemize}
\item[(B1)] Inclusivity: \((\forall\ xy )\) \(([x,y,y] \wedge [x,x,y])\)
\item[(B2)] Symmetry: \((\forall\ xzy )\) \(([x,z,y] \rightarrow [y,z,x])\)
\item[(B3)] Uniqueness: \((\forall\ xz)\) \(([x,z,x]\rightarrow x=z)\)
\end{itemize}
Given a betweenness structure \(\langle X,[\cdot,\cdot,\cdot] \rangle\), an interval is defined as \([a,b]=\{c\in X:[a,c,b]\}\), a convex subset of \(X\) is a subset \(C\) of \(X\) such that \(a,b\in C\), implies \([a,b]\subset C\). The span of a subset \(A\) of \(X\) is defined as \([A]=\bigcup\{[a,b]:a,b\in A\}\). The convex hull of \(A\) is defined as \([A]^{\omega}=\bigcup\{[A]^{n}:n\in\omega\}\), where \([A]^{0}=A\) and \([A]^{n+1}=[[A]^{n}]\).
With a detailed analysis of examples, in this paper the authors show how the notion of betweenness is related to several important concepts in mathematics. In particular if besides a betweenness structure, \(X\) has a topology, it is possible to define interesting relations between the two structures, starting by asking that intervals are closed. In this sense, the authors define local convexity, upper (and lower) semi-continuity of betweenness, and a type of internal continuity of the betweenness. They obtain results connecting the convexity and the topology in compact connected Hausdorff spaces which are aposyndetic or hereditary unicoherent. In particular, they study how the span and the convex hull interact with the topological closure and interior operators.
Reviewer: Alejandro Illanes (Ciudad de México)On the KKM theory on ordered spaceshttps://www.zbmath.org/1483.540322022-05-16T20:40:13.078697Z"Park, Sehie"https://www.zbmath.org/authors/?q=ai:park.sehieSummary: Since \textit{C. D. Horvath} and \textit{J. V. Llinares Cisar} [J. Math. Econ. 25, No. 3, 291--306 (1996; Zbl 0852.90006)] began to study maximal elements and fixed points for binary relations on topological ordered spaces, there have appeared many works related to the KKM theory on such spaces by several authors. Independently to these works, we began to study the KKM theory on abstract convex spaces [the author, Nonlinear Anal. Forum 11, No. 1, 67--77 (2006; Zbl 1120.47038)]. Our aim in the present paper is to extend the known results on topological ordered spaces to the corresponding ones on abstract convex spaces.How many simplices are needed to triangulate a Grassmannian?https://www.zbmath.org/1483.570302022-05-16T20:40:13.078697Z"Govc, Dejan"https://www.zbmath.org/authors/?q=ai:govc.dejan"Marzantowicz, Wacław"https://www.zbmath.org/authors/?q=ai:marzantowicz.waclaw"Pavešić, Petar"https://www.zbmath.org/authors/?q=ai:pavesic.petarThe aim of this article is clearly expressed in the title. The authors prove lower bounds for the number of vertices or top-dimensional simplices (facets) for a triangulation of the Grassmannian \(G_k(\mathbb{R}^n)\). The estimates are based on the knowledge of the maximal height of cup-powers in the cohomology. This method was developed by the authors [Discrete Comput. Geom. 63, No. 1, 31--48 (2020; Zbl 1431.55003)]. For the projective space \(\mathbb{R}P^{n-1}\) (the case \(k=1\)) a lower bound for the number of vertices is \(\binom{n+1}{2}\). For \(G_2(\mathbb{R}^n)\) the authors obtain \((n-2)^2+ 2^s(2n-2^s- 1)\) if \(2^s<n\le 2^{s+1}\). Similar bounds are given for \(k=3\) and \(k=4\), for \(G_3(\mathbb{R}^8)\) the bound is \(117\).
Another tool is the Lower Bound Theorem (LBT) together with its generalizations. This implies in Thm. 4.7 that the number of facets in a triangulated \(G_k(\mathbb{R}^{n+k})\) is at least \(2^{nk}+k(n^3+ n^2)/2+ k(k+2)(k-1)n/2\).
Surprisingly, this bound is sharp for \(k=1\), \(n=2\) since the smallest triangulation of \(\mathbb{R}P^2\) has \(10\) triangles. Unfortunately, for \(k\ge 2\) explicit examples do not seem to be known. So the question remains open how sharp these bounds are.
The authors remark that their method can be used also for other spaces, like Lie groups, flag manifolds, Stiefel manifolds etc. For Lie groups see the recent paper by \textit{H. Duan} et al. [Topology Appl. 293, Article ID 107559, 13 p. (2021; Zbl 1468.57022)].
Reviewer: Wolfgang Kühnel (Stuttgart)Large deviations of convex hulls of planar random walks and Brownian motionshttps://www.zbmath.org/1483.600152022-05-16T20:40:13.078697Z"Akopyan, Arseniy"https://www.zbmath.org/authors/?q=ai:akopyan.arseniy-v"Vysotsky, Vladislav"https://www.zbmath.org/authors/?q=ai:vysotsky.vladislav-vSummary: We prove large deviations principles (LDPs) for the perimeter and the area of the convex hull of a planar random walk with finite Laplace transform of its increments.
We give explicit upper and lower bounds for the rate function of the perimeter in terms of the rate function of the increments. These bounds coincide and thus give the rate function for a wide class of distributions which includes the Gaussians and the rotationally invariant ones. For random walks with such increments, large deviations of the perimeter are attained by the trajectories that asymptotically align into line segments. However, line segments may not be optimal in general.
Furthermore, we find explicitly the rate function of the area of the convex hull for random walks with rotationally invariant distribution of increments. For such walks, which necessarily have zero mean, large deviations of the area are attained by the trajectories that asymptotically align into half-circles. For random walks with non-zero mean increments, we find the rate function of the area for Gaussian walks with drift. Here the optimal limit shapes are elliptic arcs if the covariance matrix of increments is non-degenerate and parabolic arcs if otherwise.
The above results on convex hulls of Gaussian random walks remain valid for convex hulls of planar Brownian motions of all possible parameters. Moreover, we extend the LDPs for the perimeter and the area of convex hulls to general Lévy processes with finite Laplace transform.Thin-shell theory for rotationally invariant random simpliceshttps://www.zbmath.org/1483.600372022-05-16T20:40:13.078697Z"Heiny, Johannes"https://www.zbmath.org/authors/?q=ai:heiny.johannes"Johnston, Samuel"https://www.zbmath.org/authors/?q=ai:johnston.samuel-g-g"Prochno, Joscha"https://www.zbmath.org/authors/?q=ai:prochno.joschaSummary: For fixed functions \(G,H : [0,\infty)\to [0,\infty)\), consider the rotationally invariant probability density on \(\mathbb{R}^n\) of the form
\[
\mu^n (\mathrm{d}s)=\frac{1}{Z_n} G(\| s\|_2)\mathrm{e}^{-nH(\| s\|_2)}\mathrm{d}s.
\]
We show that when \(n\) is large, the Euclidean norm \(\| Y^n\|_2\) of a random vector \(Y^n\) distributed according to \(\mu^n\) satisfies a thin-shell property, in that its distribution is highly likely to concentrate around a value \(s_0\) minimizing a certain variational problem. Moreover, we show that the fluctuations of this modulus away from \(s_0\) have the order \(1/\sqrt{n}\) and are approximately Gaussian when \(n\) is large.
We apply these observations to rotationally invariant random simplices: the simplex whose vertices consist of the origin as well as independent random vectors \(Y_1^n,\ldots,Y_p^n\) distributed according to \(\mu^n\), ultimately showing that the logarithmic volume of the resulting simplex exhibits highly Gaussian behavior. Our class of measures includes the Gaussian distribution, the beta distribution and the beta prime distribution on \(\mathbb{R}^n\), provided a generalizing and unifying setting for the objects considered in [\textit{J. Grote} et al., ALEA, Lat. Am. J. Probab. Math. Stat. 16, No. 1, 141--177 (2019; Zbl 1456.52006)].
Finally, the volumes of random simplices may be related to the determinants of random matrices, and we use our methods with this correspondence to show that if \(A^n\) is an \(n\times n\) random matrix whose entries are independent standard Gaussian random variables, then there are explicit constants \(c_0, c_1\in (0,\infty)\) and an absolute constant \(C\in (0,\infty)\) such that
\[
\underset{s\in \mathbb{R}}{\sup} \left| \mathbb{P}\left[ \dfrac{\log \mathrm{det} (A^n) -\log (n-1)\! -c_0}{\sqrt{\frac{1}{2}\log n+c_1}} < s\right] -\int_{-\infty}^s \dfrac{\mathrm{e}^{-u^2 /2} \mathrm{d}u}{\sqrt{2\pi}} \right| < \dfrac{C}{\log^{3/2}n},
\]
sharpening the \(1/\log^{1/3+o(1)} n\) bound in [\textit{H. H. Nguyen} and \textit{V. Vu}, Ann. Probab. 42, No. 1, 146--167 (2014; Zbl 1299.60005)].Free-boundary conformal parameterization of point cloudshttps://www.zbmath.org/1483.650322022-05-16T20:40:13.078697Z"Choi, Gary P. T."https://www.zbmath.org/authors/?q=ai:choi.gary-pui-tung"Liu, Yechen"https://www.zbmath.org/authors/?q=ai:liu.yechen"Lui, Lok Ming"https://www.zbmath.org/authors/?q=ai:lui.lok-mingSummary: With the advancement in 3D scanning technology, there has been a surge of interest in the use of point clouds in science and engineering. To facilitate the computations and analyses of point clouds, prior works have considered parameterizing them onto some simple planar domains with a fixed boundary shape such as a unit circle or a rectangle. However, the geometry of the fixed shape may lead to some undesirable distortion in the parameterization. It is therefore more natural to consider free-boundary conformal parameterizations of point clouds, which minimize the local geometric distortion of the mapping without constraining the overall shape. In this work, we develop a free-boundary conformal parameterization method for disk-type point clouds, which involves a novel approximation scheme of the point cloud Laplacian with accumulated cotangent weights together with a special treatment at the boundary points. With the aid of the free-boundary conformal parameterization, high-quality point cloud meshing can be easily achieved. Furthermore, we show that using the idea of conformal welding in complex analysis, the point cloud conformal parameterization can be computed in a divide-and-conquer manner. Experimental results are presented to demonstrate the effectiveness of the proposed method.Unlabeled sample compression schemes and corner peelings for ample and maximum classeshttps://www.zbmath.org/1483.682812022-05-16T20:40:13.078697Z"Chalopin, Jérémie"https://www.zbmath.org/authors/?q=ai:chalopin.jeremie"Chepoi, Victor"https://www.zbmath.org/authors/?q=ai:chepoi.victor-d"Moran, Shay"https://www.zbmath.org/authors/?q=ai:moran.shay"Warmuth, Manfred K."https://www.zbmath.org/authors/?q=ai:warmuth.manfred-kSummary: We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by
\textit{H. Tracy Hall} [Counterexamples in discrete geometry. University of California (PhD thesis) (2004)]
of a partial shelling of the cross-polytope which can not be extended. From it, we derive a maximum class of VC dimension 3 without corners. This refutes several previous works in machine learning. In particular, it implies that the previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an optimal unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample classes, which generalize maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the associated 1-inclusion graph.A filtering heuristic for the computation of minimum-volume enclosing ellipsoidshttps://www.zbmath.org/1483.684642022-05-16T20:40:13.078697Z"Källberg, Linus"https://www.zbmath.org/authors/?q=ai:kallberg.linus"Larsson, Thomas"https://www.zbmath.org/authors/?q=ai:larsson.thomasSummary: We study heuristics to accelerate existing state-of-the-art algorithms for the minimum-volume enclosing ellipsoid problem. We propose a new filtering heuristic that can significantly reduce the number of distance computations performed in algorithms derived from Khachiyan's first-order algorithm. Our experiments indicate that in high dimensions, the filtering heuristic is more effective than the elimination heuristic proposed by Harman and Pronzato. In lower dimensions, the elimination heuristic is superior.
For the entire collection see [Zbl 1377.68004].Algorithms for colourful simplicial depth and medians in the planehttps://www.zbmath.org/1483.684722022-05-16T20:40:13.078697Z"Zasenko, Olga"https://www.zbmath.org/authors/?q=ai:zasenko.olga"Stephen, Tamon"https://www.zbmath.org/authors/?q=ai:stephen.tamonSummary: The colourful simplicial depth (CSD) of a point \(x \in \mathbb {R}^2\) relative to a configuration \(P=(P^1, P^2, \ldots , P^k)\) of \(n\) points in \(k\) colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain \(x\) in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point \(x\), and of finding a point in \(x \in \mathbb {R}^2\), called a median, that maximizes colourful simplicial depth.
For computing the colourful simplicial depth of \(x\), our algorithm runs in time \(O\left( n \log {n} + k n \right) \) in general, and \(O\)(kn) if the points are sorted around \(x\). For finding the colourful median, we get a time of \(O(n^4)\). For comparison, the running times of the best known algorithm for the monochrome version of these problems are \(O\left( n \log {n} \right) \) in general, improving to \(O(n)\) if the points are sorted around \(x\) for monochrome depth, and \(O(n^4)\) for finding a monochrome median.
For the entire collection see [Zbl 1377.68004].A study of convex convex-composite functions via infimal convolution with applicationshttps://www.zbmath.org/1483.901092022-05-16T20:40:13.078697Z"Burke, James V."https://www.zbmath.org/authors/?q=ai:burke.james-v"Hoheisel, Tim"https://www.zbmath.org/authors/?q=ai:hoheisel.tim"Nguyen, Quang V."https://www.zbmath.org/authors/?q=ai:van-nguyen.quangThe authors provide a fresh look on conjugacy and subdifferential calculus for convex composed functions defined on finitely dimensional spaces by means of relaxed monotonicity assumptions that are weaker than in the previous literature. The theoretical achievements are illustrated by several applications in optimization and matrix analysis, problems from conic programming and involving matrix-fractional, variational Gram, and spectral functions finally becoming approachable via the new framework.
Reviewer: Sorin-Mihai Grad (Paris)