Pairs of compact convex sets. Fractional arithmetic with convex sets.

*(English)*Zbl 1027.46001
Mathematics and its Applications (Dordrecht). 548. Dordrecht: Kluwer Academic Publishers. xii, 295 p. (2002).

Let \(X=\{X,\tau\}\) be a real topological vector space and \(\mathcal B(X)\) the set of all nonempty bounded closed convex subsets of \(X\). The present book is devoted to the theory of pairs of compact convex sets \((A,B)\in\mathcal B^2(X)(=\mathcal B(X)\times\mathcal B(X))\) and in particular to the problem of finding different types of minimal representatives of a pair of nonempty compact convex subsets of a locally convex vector space in the sense of the Rådström-Hörmander theory. Special attention is given to the two-dimensional case, where the minimal pairs are uniquely determined up to translations. This fact is not true in higher dimensional spaces and leads to a beautiful theory on the mutual interactions between minimality under constraints, separation and decomposition of convex sets, convexificators and invariants of minimal pairs.

This theory sheds light upon both sides of the collection of all compact convex subsets of a locally vector space, namely the geometric and the algebraic one. From the algebraic point of view, the collection of all nonempty compact convex subsets of a topological vector space is an ordered semigroup with cancellation property under the inclusion of sets and the Minkowski-addition. In this approach, pairs of nonempty compact convex sets correspond to fractions of elements from the semigroup and minimal pairs to relatively prime fractions. Since the pairs of compact convex sets are not uniquely determined, minimal representations are of special importance. A problem related to the existence of minimal pairs of compact convex sets is the existence of reduced pairs of convex bodies. The main difficulty in finding geometric or algebraic conditions for minimality is the lack of any theory which could be used to determine minimal pairs. Therefore, the authors concentrate on two typical examples of pairs of compact convex sets in the plane to clarify the difficulties: The pair of orthogonal lenses and the Star of David considered as pair of two equilateral triangles which have the same center and are placed in opposite directions. Both examples turn out to be typical of minimality and became the starting point of an extensive theory whose geometric and algebraic beauty is the topic of this book.

In Chapters 1-3, Convex Sets and Sublinearity, Topological Vector Spaces, Compact Convex Sets, the authors give some preliminary remarks about ordering and give a short survey about the basic facts on sublinear functions. The algebraic version of the Hahn-Banach theorem and the separation theorem for convex sets in vector spaces and for locally convex vector spaces are proved. The basic facts about convexity, mixed volumes and the Rådström-Hörmander lattice are presented. In Chapter 4, Minimal Pairs of Convex Sets, the elementary properties of minimal pairs of compact convex subsets in topological vector spaces are investigated. The existence of minimal representatives in every equivalence class of pairs of compact convex sets is proved. In general, no minimal representatives exist for pairs of bounded closed convex sets. Several necessary and sufficient conditions for the minimality of pairs of compact convex sets are proved. In Chapter 5, The Cardinality of Minimal Pairs, it is shown that minimal pairs are not uniquely determined. However, in some special cases the uniqueness or at least some uniqueness properties are proved. The authors begin with pairs of bounded closed convex sets which lie in complementary subspaces. Then minimal pairs of compact convex subsets in the plane are considered.

In Chapter 6, Minimality under Constraints, the minimality of pairs of closed bounded convex sets under constraints is studied. For example, pairs \((C,D)\in\mathcal B^2(X)\) with convex union \(C\cup D\in\mathcal B(X)\) are investigated and the equivalent minimal pairs with convex union are determined. In Chapter 7, Symmetries, symmetric pairs of convex compact sets are discussed and the minimality in asymmetry classes of convex compact sets with respect to inclusion is studied. It is shown that there is a connection between asymmetry classes and the Pinsker-Minkowski-Rådström-Hörmander lattice. Minimal pairs of compact convex sets can be constructed by decomposing a given compact convex set. In Chapter 8, Decompositions are performed in two different ways. One way is to determine nontrivial summands, another way is to divide the convex set by a hyperplane into two nonempty convex subsets. For pairs of compact convex sets there exist several invariants. One of these invariants is discussed in Chapter 3, namely the steepest ascent and descent directions of the corresponding DCH-function (the function from the real vector space of differences of continuous sublinear functions).

In Chapter 9, Invariants, which belong only to minimal pairs of compact convex sets and not belong to the whole class are discussed. Such invariants are the affine dimension and codimension of the union of a minimal pair of compact convex sets. Pairs of compact convex sets arise in the quasidifferential calculus of V. F. Demyanov and A. M. Rubinov as sub- and superdifferentials of quasidifferentiable functions, in non-smooth analysis, in set-valued analysis, in the field of combinatorial convexity and in the formulas for the numerical evaluation of the Aumann integral. In Chapter 10, Applications to the piecewise differentiable functions, DCH-functions, quasidifferentiable functions, strong directional derivatives of set-valued functions and quadrature formulas for the Aumann integral are given. The set \(\mathcal B(X)\) of all nonempty bounded convex subsets of a topological vector space \(X\) endowed with the multiplication \(\ast\) given by \(A\ast B = A \dotplus B\) and the ordering induced by the inclusion is an ordered commutative semigroup which satisfies the order cancellation law. The ordering as well as the multiplication can be extended to the equivalence class \(\mathcal B^2(X)/\sim\) which contains \((A,B)\in\mathcal B^2\) by \[ [A,B]\preceq[C,D] \Longleftrightarrow A \dotplus D\subseteq B \dotplus C \quad\text{and}\quad [A, B] \star [C, D] = [A\ast C,B\ast D]. \] Within this context the elements of \(\mathcal B^2(X)\sim\) can be considered as fractions of elements of \(\mathcal B(X)\). The definition of the ordering and the multiplication does not depend on the special choice of the representatives.

In Chapter 11, Fractions, the problem of finding minimal representatives for the elements of \(\mathcal B^2(X)/\sim\) as a special case of the general problem of determining minimal fractions in ordered commutative semigroups which satisfy the order cancellation law is considered. In Chapter 12, Piecewise Linear Functions, a very brief introduction to some basic concepts from combinatorial convexity is given. The theorem on a basis for the Picard group of the coarset fan \(\Sigma_n\), Pic\((\Sigma_n)\), is proved. A constructive proof of this theorem for the two-dimensional case is given. In the case \(\Sigma_3\), the representation theorem is stated.

This theory sheds light upon both sides of the collection of all compact convex subsets of a locally vector space, namely the geometric and the algebraic one. From the algebraic point of view, the collection of all nonempty compact convex subsets of a topological vector space is an ordered semigroup with cancellation property under the inclusion of sets and the Minkowski-addition. In this approach, pairs of nonempty compact convex sets correspond to fractions of elements from the semigroup and minimal pairs to relatively prime fractions. Since the pairs of compact convex sets are not uniquely determined, minimal representations are of special importance. A problem related to the existence of minimal pairs of compact convex sets is the existence of reduced pairs of convex bodies. The main difficulty in finding geometric or algebraic conditions for minimality is the lack of any theory which could be used to determine minimal pairs. Therefore, the authors concentrate on two typical examples of pairs of compact convex sets in the plane to clarify the difficulties: The pair of orthogonal lenses and the Star of David considered as pair of two equilateral triangles which have the same center and are placed in opposite directions. Both examples turn out to be typical of minimality and became the starting point of an extensive theory whose geometric and algebraic beauty is the topic of this book.

In Chapters 1-3, Convex Sets and Sublinearity, Topological Vector Spaces, Compact Convex Sets, the authors give some preliminary remarks about ordering and give a short survey about the basic facts on sublinear functions. The algebraic version of the Hahn-Banach theorem and the separation theorem for convex sets in vector spaces and for locally convex vector spaces are proved. The basic facts about convexity, mixed volumes and the Rådström-Hörmander lattice are presented. In Chapter 4, Minimal Pairs of Convex Sets, the elementary properties of minimal pairs of compact convex subsets in topological vector spaces are investigated. The existence of minimal representatives in every equivalence class of pairs of compact convex sets is proved. In general, no minimal representatives exist for pairs of bounded closed convex sets. Several necessary and sufficient conditions for the minimality of pairs of compact convex sets are proved. In Chapter 5, The Cardinality of Minimal Pairs, it is shown that minimal pairs are not uniquely determined. However, in some special cases the uniqueness or at least some uniqueness properties are proved. The authors begin with pairs of bounded closed convex sets which lie in complementary subspaces. Then minimal pairs of compact convex subsets in the plane are considered.

In Chapter 6, Minimality under Constraints, the minimality of pairs of closed bounded convex sets under constraints is studied. For example, pairs \((C,D)\in\mathcal B^2(X)\) with convex union \(C\cup D\in\mathcal B(X)\) are investigated and the equivalent minimal pairs with convex union are determined. In Chapter 7, Symmetries, symmetric pairs of convex compact sets are discussed and the minimality in asymmetry classes of convex compact sets with respect to inclusion is studied. It is shown that there is a connection between asymmetry classes and the Pinsker-Minkowski-Rådström-Hörmander lattice. Minimal pairs of compact convex sets can be constructed by decomposing a given compact convex set. In Chapter 8, Decompositions are performed in two different ways. One way is to determine nontrivial summands, another way is to divide the convex set by a hyperplane into two nonempty convex subsets. For pairs of compact convex sets there exist several invariants. One of these invariants is discussed in Chapter 3, namely the steepest ascent and descent directions of the corresponding DCH-function (the function from the real vector space of differences of continuous sublinear functions).

In Chapter 9, Invariants, which belong only to minimal pairs of compact convex sets and not belong to the whole class are discussed. Such invariants are the affine dimension and codimension of the union of a minimal pair of compact convex sets. Pairs of compact convex sets arise in the quasidifferential calculus of V. F. Demyanov and A. M. Rubinov as sub- and superdifferentials of quasidifferentiable functions, in non-smooth analysis, in set-valued analysis, in the field of combinatorial convexity and in the formulas for the numerical evaluation of the Aumann integral. In Chapter 10, Applications to the piecewise differentiable functions, DCH-functions, quasidifferentiable functions, strong directional derivatives of set-valued functions and quadrature formulas for the Aumann integral are given. The set \(\mathcal B(X)\) of all nonempty bounded convex subsets of a topological vector space \(X\) endowed with the multiplication \(\ast\) given by \(A\ast B = A \dotplus B\) and the ordering induced by the inclusion is an ordered commutative semigroup which satisfies the order cancellation law. The ordering as well as the multiplication can be extended to the equivalence class \(\mathcal B^2(X)/\sim\) which contains \((A,B)\in\mathcal B^2\) by \[ [A,B]\preceq[C,D] \Longleftrightarrow A \dotplus D\subseteq B \dotplus C \quad\text{and}\quad [A, B] \star [C, D] = [A\ast C,B\ast D]. \] Within this context the elements of \(\mathcal B^2(X)\sim\) can be considered as fractions of elements of \(\mathcal B(X)\). The definition of the ordering and the multiplication does not depend on the special choice of the representatives.

In Chapter 11, Fractions, the problem of finding minimal representatives for the elements of \(\mathcal B^2(X)/\sim\) as a special case of the general problem of determining minimal fractions in ordered commutative semigroups which satisfy the order cancellation law is considered. In Chapter 12, Piecewise Linear Functions, a very brief introduction to some basic concepts from combinatorial convexity is given. The theorem on a basis for the Picard group of the coarset fan \(\Sigma_n\), Pic\((\Sigma_n)\), is proved. A constructive proof of this theorem for the two-dimensional case is given. In the case \(\Sigma_3\), the representation theorem is stated.

Reviewer: Serguey M.Pokas (Odessa)

##### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

46A55 | Convex sets in topological linear spaces; Choquet theory |