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Mat. Zametki, 2008, Volume 84, Issue 4, Pages 496–505 (Mi mz4093)  

This article is cited in 1 scientific paper (total in 1 paper)

Properties of $P$-sets and Trapped Compact Convex Sets

M. V. Balashov, I. I. Bogdanov

Moscow Institute of Physics and Technology

Abstract: New properties of $P$-sets, which constitute a large class of convex compact sets in $\mathbb R^n$ 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 $\mathbb R^n$ 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 $M\subset\mathbb R^n$, there exists a set $A^0\subset A$ trapped in $M$ and containing at most $2n$ elements. An example of a convex compact set $M\subset\mathbb R^n$ for which such a finite set $A^0\subset A$ does not exist is given.

Keywords: set-valued map, compact convex set, $P$-set, trapped set, selector, Hausdorff metric, upper (lower) semicontinuous map, Lipschitz continuity

DOI: https://doi.org/10.4213/mzm4093

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English version:
Mathematical Notes, 2008, 84:4, 465–472

Bibliographic databases:

UDC: 517.98
Received: 02.03.2005
Revised: 15.02.2007

Citation: M. V. Balashov, I. I. Bogdanov, “Properties of $P$-sets and Trapped Compact Convex Sets”, Mat. Zametki, 84:4 (2008), 496–505; Math. Notes, 84:4 (2008), 465–472

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. M. I. Gomoyunov, N. Yu. Lukoyanov, “On the stability of a procedure for solving a minimax control problem for a positional functional”, Proc. Steklov Inst. Math. (Suppl.), 288, suppl. 1 (2015), 54–69  mathnet  crossref  mathscinet  isi  elib
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