RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Mat. Zametki: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (451 kB)
References: PDF file   HTML file

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

Bibliographic databases:

UDC: 517.98
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
\Bibitem{BalBog08} \by M.~V.~Balashov, I.~I.~Bogdanov \paper Properties of $P$-sets and Trapped Compact Convex Sets \jour Mat. Zametki \yr 2008 \vol 84 \issue 4 \pages 496--505 \mathnet{http://mi.mathnet.ru/mz4093} \crossref{https://doi.org/10.4213/mzm4093} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2485190} \zmath{https://zbmath.org/?q=an:1155.52301} \transl \jour Math. Notes \yr 2008 \vol 84 \issue 4 \pages 465--472 \crossref{https://doi.org/10.1134/S0001434608090186} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000260516700018} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-55149093047} 

• http://mi.mathnet.ru/eng/mz4093
• https://doi.org/10.4213/mzm4093
• http://mi.mathnet.ru/eng/mz/v84/i4/p496

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
•  Number of views: This page: 422 Full text: 127 References: 47 First page: 8