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Journal of Convex Analysis 14 (2007), No. 3, 565--588
Copyright Heldermann Verlag 2007

Abstract Convexity and Connectedness

Jürgen Kindler
Fachbereich Mathematik, Technische Universität, Schlossgartenstr. 7, 64289 Darmstadt, Germany

In a former paper the concept of n-ary connectedness was introduced, where 1-ary connectedness coincides with the usual notion of (abstract) connectedness. In the present paper, sets endowed with a convexity structure are studied, where the polytopes are n-ary connected. Interrelations between the classical Helly and Carathéodory numbers are evaluated as a pre-stage of Helly and Carathéodory type intersection theorems. Various other applications such as intersection theorems and fixed point theorems for trees and hyperconvex metric spaces are presented.

Keywords: Connected, convex, star-shaped, selector, set-valued function.

MSC: 52A01, 52A30, 52A35, 54D05

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