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

Editorial: Springer, 2010

Idioma: Ingl�s

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Detalles bibliogr�ficos

T�tulo

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

Editor

Springer

Fecha de publicaci�n

2010

Idioma

Ingl�s

ISBN 10

9048161495

ISBN 13

9789048161492

Encuadernaci�n

Encuadernaci�n de tapa blanda

Estado

New

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Sinopsis

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 R�dstr�m-H�rmander 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 R�dstr�m-H�rmander 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.

Rese�a 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]).

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