Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 2, pp. 441449 (2003) 

An analogue of the KreinMilman theorem for starshaped setsHorst Martini and Walter WenzelFaculty of Mathematics, University of Technology Chemnitz, D09107 Chemnitz, GermanyAbstract: Motivated by typical questions from computational geometry (visibility and art gallery problems) and combinatorial geometry (illumination problems) we present an analogue of the KreinMilman theorem for the class of starshaped sets. If $S\subseteq\mathbb{R}^n$ is compact and starshaped, we consider a fixed, nonempty, compact, and convex subset $K$ of the convex kernel $K_0=\mbox{ck}(S)\mbox{ of }S$, for instance $K=K_0$ itself. A point $q_0\in S\setminus K$ will be called an extreme point of $S$ modulo $K$, if for all $p\in S\setminus(K\cup\{q_0\})$ the convex closure of $K\cup\{p\}$ does not contain $q_0$. We study a closure operator $\sigma:{\cal P}(\mathbb{R}^n\setminus K)\longrightarrow{\cal P} (\bR^n\setminus K)$ induced by visibility problems and prove that $\sigma(S_0)=S\setminus K$, where $S_0$ denotes the set of extreme points of $S$ modulo $K$. Keywords: convex sets, starshaped sets, closure operators, KreinMilman theorem, visibility problems, illumination problems, watchman route problem, $d$dimensional volume Classification (MSC2000): 52A30; 06A15, 5201, 52A20, 52A43 Full text of the article:
Electronic version published on: 1 Aug 2003. This page was last modified: 4 May 2006.
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