Valuations and polarity

Abstract

Given a collection[Figure not available: see fulltext.] of convex polytopes, let τ([Figure not available: see fulltext.]) denote the set of all convex transversals of[Figure not available: see fulltext.]. If[Figure not available: see fulltext.] and ℬ are two such collections, of finite cardinality, then there is a simple, arithmetical condition which holds precisely when τ([Figure not available: see fulltext.])=τ(ℬ). Another such condition, involving what we call the "Sallee-Shephard mapping," characterizes those pairs[Figure not available: see fulltext.] and ℬ for which τ(τ([Figure not available: see fulltext.]))=τ(ℬ). As these results are established, several distributive lattices involving convex sets are introduced, and relationships between their valuation modules are determined. In particular, it is proven that the Sallee-Shephard mapping is an isomorphism of the additive, abelian group of simple functions generated by the characteristic functions of the open, convex sets and that generated by those of the closed, convex sets. © 1988 Springer-Verlag New York Inc.