Pairs of Compact Convex Sets
Fractional Arithmetic With Convex Sets
- Editore:
Springer Netherlands
- Collana:
- Mathematics And Its Applications
- EAN:
9781402009389
- ISBN:
1402009380
- Pagine:
- 312
- Formato:
- Hardback
- Lingua:
- Inglese
Descrizione Pairs of Compact Convex Sets
The book is devoted to the theory of pairs of compact convexsets and in particular to the problem of finding different types ofminimal representants of a pair of nonempty compact convex subsets ofa locally convex vector space in the sense of theRadstrom-Hormander Theory. Minimal pairs of compactconvex sets arise naturally in different fields of mathematics, as forinstance in non-smooth analysis, set-valued analysis and in the fieldof combinatorial convexity.In the first three chapters of the book the basic facts aboutconvexity, mixed volumes and the Radstrom-Hormanderlattice are presented. Then, a comprehensive theory oninclusion-minimal representants of pairs of compact convex sets isgiven. Special attention is given to the two-dimensional case, wherethe minimal pairs are uniquely determined up to translations. Thisfact is not true in higher dimensional spaces and leads to a beautifultheory on the mutual interactions between minimality underconstraints, separation and decomposition of convex sets, convexificators and invariants of minimal pairs. This theory throwslight upon both sides of the collection of all compact convex subsetsof a locally vector space, namely the geometric and the algebraicone.From the algebraic point of view the collection of all nonemptycompact convex subsets of a topological vector space is an orderedsemi group with cancellation property under the inclusion of sets andthe Minkowski-addition. From this approach pairs of nonempty compactconvex sets correspond to fractions of elements from the semi groupand minimal pairs to relatively prime fractions.
