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

Editore: Springer, 2010

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Dati bibliografici

• Titolo: Pairs of Compact Convex Sets: Fractional Arithmetic with Convex Sets (Mathematics and Its Applications)
• Editore: Springer
• Data di pubblicazione: 2010
• Lingua: Inglese
• ISBN 10: 9048161495
• ISBN 13: 9789048161492
• Rilegatura: Brossura
• Condizione: New

Descrizione articolo

Riassunto

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.

Contenuti

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.