Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Keywords

  • Connected
  • convex
  • star-shaped
  • selector
  • set-valued function

Mathematics Subject Classification

52A01, 52A30, 52A35, 54D05

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Suggested citation

J. Kindler. “Abstract Convexity and Connectedness.” Journal of Convex Analysis 14 (2007), No. 3, 565–588. https://doi.org/10.68381/jca14034

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Published by Heldermann Verlag, 2007. Rights now held by Banach Press LLC.