Recent zbMATH articles in MSC 37J11https://www.zbmath.org/atom/cc/37J112022-04-28T18:15:44.554837ZWerkzeugCorrigendum and addendum to: ``Computation of domains of analyticity for the dissipative standard map in the limit of small dissipation''https://www.zbmath.org/1482.370522022-04-28T18:15:44.554837Z"Bustamante, Adrián P."https://www.zbmath.org/authors/?q=ai:bustamante.adrian-p"Calleja, Renato C."https://www.zbmath.org/authors/?q=ai:calleja.renato-cSummary: We correct some tables and figures in our paper [ibid. 395, 15--23 (2019; Zbl 1451.37080)]. We also report on the new computations that verify the accuracy of the data and extend the results. The new computations have led us to find new patterns in the data that were not noticed before. We formulate some more precise conjectures.Duality of gauges and symplectic forms in vector spaceshttps://www.zbmath.org/1482.520102022-04-28T18:15:44.554837Z"Balestro, Vitor"https://www.zbmath.org/authors/?q=ai:balestro.vitor"Martini, Horst"https://www.zbmath.org/authors/?q=ai:martini.horst"Teixeira, Ralph"https://www.zbmath.org/authors/?q=ai:teixeira.ralph-costaLet \(X\) be a finite dimensional vector space endowed with the unique Hausdorff vector topology. By a convex body one understands a compact convex subset \(K\) of \(X\) with \(0\in \operatorname{int}K\). The set of all these convex bodies is denoted by \(\mathcal{K}_0(X)\). The Minkowski functional \(\gamma_K\), given by \(\gamma_K(x)=\inf\{\lambda\ge 0: x\in \lambda K\}\), is called the gauge associated to \(K\). Then \(\gamma_K\) is an asymmetric norm on \(X\), i.e. a positively definite, positively homogeneous and subadditive functional on \(X\). Conversely, every asymmetric norm \(\gamma\) on \(X\) is the gauge of its unit ball \(B_\gamma=\{x\in X:\gamma(x)\le 1\}\). The topology generated by \(\gamma\) agrees with the initial topology of \(X\). The dual gauge of \(\gamma\) is the functional \(\gamma^*\) defined on the dual space \(X^*\) of \(X\) by \(\gamma^*(f)=\max\{f(x): x\in K\}\) and the polar body of \(K\) is \(K^\circ =\{f\in X^*: f(x)\le 1,\, \forall x\in K\}\). It turns out that \(\gamma^*\) is a gauge on \(X^*\), namely the gauge asoociated to the polar body \(K^\circ\).
Suppose now that \(X\) is even-dimensional and let \(\omega:X\times X\to\mathbb{R}\) be a fixed symplectic form on \(X\), i.e. a nondegenerate alternating bilinear form on \(X\). Then \(\omega\) yields an identification of \(X\) and its dual space \(X^*\). Namely, for every \(f\in X^*\) there is a unique \(x_f\in X\) s.t. \(f(\cdot)=\omega(x_f,\cdot)\) and the mapping \(\mathcal I:X^*\to X\) given by \(\mathcal I(f)=x_f\) is an isomorphism. If \(K\in\mathcal{K}_0(X)\), then \(K^\omega:=\mathcal I(K^\circ)\) is called the dual body of \(K\). It follows that \(K^\omega\in\mathcal{K}_0(X)\), the gauge \(\gamma_{K^\omega}\) satisfies the equality \(\gamma_{K^\omega}(x)=\max\{\omega(x,y):y\in K\}\) and its dual gauge is \(\gamma_{-K}\).
As the authors say in the Abstract: ``In this paper, we study geometric properties of this so-called dual gauge, such as its behavior under isometries and its relation to orthogonality. A version of the Mazur-Ulam theorem for gauges is also proved. As an application of the theory, we show that closed characteristics of the boundary of a (smooth) convex body are optimal cases of a certain isoperimetric inequality.''
Reviewer: Stefan Cobzaş (Cluj-Napoca)