Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 1, pp. 153-169 (2002)

On the Characterization of some Families of Closed Convex Sets

Miguel A. Goberna, Valentin Jornet, Margarita Rodriguez

Department of Statistics and Operations Research, Faculty of Sciences, University of Alicante, E-03080 Alicante, Spain, e-mail:

Abstract: This paper deals with the characterization of the sums of compact convex sets with linear subspaces, simplices, sandwiches (convex hulls of pairs of parallel affine manifolds) and parallelotopes in terms of the so-called internal and conical representations, topological and geometrical properties. In particular, it is shown that a closed convex set is a sandwich if and only if its relative boundary is unconnected. The characterizations of families of closed convex sets can be useful in different fields of applied mathematics. For instance, it is proved that a bounded linear semi-infinite programming problem whose feasible set is the sum of a compact convex set with a linear subspace is necessarily solvable and has zero duality gap.

Keywords: closed convex sets, simplices, sandwiches, parallelotopes, linear inequalities, connectivity

Classification (MSC2000): 52A20, 52A40, 52A41

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