Pairs of Compact Convex Sets: Fractional Arithmetic with Convex Sets

ISBN-13: 9789048161492 / Angielski / Miękka / 2010 / 295 str.

Diethard Ernst Pallaschke; R. Urbanski

Opis

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 Radstrom-Hormander 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 Radstrom-Hormander 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. This theory throws 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 semi group with cancellation property under the inclusion of sets and the Minkowski-addition. From this approach pairs of nonempty compact convex sets correspond to fractions of elements from the semi group and minimal pairs to relatively prime fractions."

Spis treści

Preface. I: Convexity. 1. Convex Sets and Sublinearity. 2. Topological Vector Spaces. 3. Compact Convex Sets. II: Minimal Pairs. 4. Minimal Pairs of Convex Sets. 5. The Cardinality of Minimal Pairs. 6. Minimality under Constraints. 7. Symmetries. 8. Decompositions. 9. Invariants. 10. Applications. III: Semigroups. 11. Fractions. 12. Piecewise Linear Functions. Open Questions. List of Symbols. Index. Bibliography.