Properties of P-sets and trapped compact convex sets

Abstract

New properties of P-sets, which constitute a large class of convex compact sets in ℝⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ℝⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ⊂ ℝⁿ, there exists a set A ⁰ ⊂ A trapped in M and containing at most 2n elements. An example of a convex compact set M ⊂ ℝⁿ for which such a finite set A ⁰ ⊂ A does not exist is given.