NDLI: Properties of P-sets and trapped compact convex sets

Content Provider	Springer Nature Link
Author	Balashov, M. V. Bogdav, I. I.
Copyright Year	2008
Abstract	New properties of P-sets, which constitute a large class of convex compact sets in 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 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 ⊂ ℝ$^{ n }$, there exists a set A $^{0}$ ⊂ A trapped in M and containing at most 2n elements. An example of a convex compact set M ⊂ ℝ$^{ n }$ for which such a finite set A $^{0}$ ⊂ A does not exist is given.
Starting Page	465
Ending Page	472
Page Count	8
File Format	PDF
ISSN	00014346
Journal	Mathematical Notes
Volume Number	84
Issue Number	3-4
e-ISSN	15738876
Language	English
Publisher	SP MAIK Nauka/Interperiodica
Publisher Date	2008-10-29
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	set-valued map compact convex set P-set trapped set selector Hausdorff metric Upper (lower) semicontinuous map Lipschitz continuity Mathematics
Content Type	Text
Resource Type	Article

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