
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 setvalued maps. It is also shown that if the graph of a setvalued 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 socalled trapped sets are also studied; wellknown 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:
setvalued 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 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
\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 496505
\mathnet{http://mi.mathnet.ru/mz4093}
\crossref{https://doi.org/10.4213/mzm4093}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2485190}
\zmath{https://zbmath.org/?q=an:1155.52301}
\transl
\jour Math. Notes
\yr 2008
\vol 84
\issue 4
\pages 465472
\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=2s2.055149093047}
Linking options:
http://mi.mathnet.ru/eng/mz4093https://doi.org/10.4213/mzm4093 http://mi.mathnet.ru/eng/mz/v84/i4/p496
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:

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 
