Martini, Horst ; Wenzel, Walter : An analogue of the Krein-Milman Theorem for star-shaped sets

Author(s):

Martini, Horst
Wenzel, Walter

Title:

An analogue of the Krein-Milman Theorem for star-shaped sets

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 4, 2001

Mathematics Subject Classification:

06A15			[ Galois correspondences, closure operators ]
52-01			[ Instructional exposition (textbooks, tutorial papers, etc.) ]
52A20			[ Convex sets in $n$ dimensions (including convex hypersurfaces) ]
52A30			[ Variants of convex sets (star-shaped, ($m, n$)-convex, etc.) ]

Abstract:

If $S \subseteq R^n$ is compact and star-shaped, we consider a fixed, nonempty, compact and convex subset K of the convex kernel of S. We prove that $\sigma(sol=S\backslash K$, where $\sigma : \rho(R^n \backslash K) \rightarrow \rho(R^n \backslash T)$ denotes same closure operator induced by visibility problems, and so denotes the - appropriately defined - set of extreme points of S modula K

Keywords:

convex sets, star-shaped sets, closure operators, Krein-Milman theorem, visibility problems, d- dimensional volume

Language:

English

Publication time:

11 / 2001