Pairs of Compact Convex Sets: Fractional Arithmetic with Convex Sets (Mathematics and Its Applications)

Pallaschke, Diethard Ernst; Urbanski, R.

ISBN 10: 9048161495 / ISBN 13: 9789048161492

Verlag: Springer, 2010

Beschreibung

Bestandsnummer des Verkäufers I-9789048161492

Bibliografische Details

• Titel: Pairs of Compact Convex Sets: Fractional Arithmetic with Convex Sets (Mathematics and Its Applications)
• Verlag: Springer
• Erscheinungsjahr: 2010
• Sprache: Englisch
• ISBN-10: 9048161495
• ISBN-13: 9789048161492
• Einband: Softcover
• Zustand: New

Über diesen Titel

Inhaltsangabe

The book is devoted to the theory of pairs of compact convex sets and in particular to the problem of finding different types of minimal representants of a pair of nonempty compact convex subsets of a locally convex vector space in the sense of the Rådström-Hörmander Theory. Minimal pairs of compact convex sets arise naturally in different fields of mathematics, as for instance in non-smooth analysis, set-valued analysis and in the field of combinatorial convexity. In the first three chapters of the book the basic facts about convexity, mixed volumes and the Rådström-Hörmander lattice are presented. Then, a comprehensive theory on inclusion-minimal representants of pairs of compact convex sets is given. 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.

Reseña del editor

Pairs of compact convex sets arise in the quasidifferential calculus of V.F. Demyanov and A.M. Rubinov as sub- and superdifferentials of quasidifferen­ tiable functions (see [26]) and in the formulas for the numerical evaluation of the Aumann-Integral which were recently introduced in a series of papers by R. Baier and F. Lempio (see [4], [5], [10] and [9]) and R. Baier and E.M. Farkhi [6], [7], [8]. In the field of combinatorial convexity G. Ewald et al. [36] used an interesting construction called virtual polytope, which can also be represented as a pair of polytopes for the calculation of the combinatorial Picard group of a fan. Since in all mentioned cases the pairs of compact con­ vex sets are not uniquely determined, minimal representations are of special to the existence of minimal pairs of compact importance. A problem related convex sets is the existence of reduced pairs of convex bodies, which has been studied by Chr. Bauer (see [14]).