An Analogue of the Krein-Milman Theorem for Star-Shaped Sets Dedicated to Professor Bernd Silbermann on the occasion of his 60th birthday

Motivated by typical questions from computational geometry (visibility and art gallery problems) and combinatorial geometry (illumination problems) we present an analogue of the Krein-Milman theorem for the class of star-shaped sets. If S ⊆ R is compact and star-shaped, we consider a fixed, nonempty, compact, and convex subset K of the convex kernel K0 = ck(S) of S, for instance K = K0 itself. A point q0 ∈ S \K will be called an extreme point of S modulo K, if for all p ∈ S \ (K ∪ {q0}) the convex closure of K ∪ {p} does not contain q0. We study a closure operator σ : P(R \ K) −→ P(R \ K) induced by visibility problems and prove that σ(S0) = S \ K, where S0 denotes the set of extreme points of S modulo K. MSC 2000: 52A30 (primary); 06A15, 52-01, 52A20, 52A43 (secondary)