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# \v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$

Published:2009-04-01
Printed: Apr 2009
• Robert J. MacG. Dawson
• Maria Moszy\'{n}ska
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## Abstract

A set in a metric space is called a \emph{\v{C}eby\v{s}ev set} if it has a unique nearest neighbour'' to each point of the space. In this paper we generalize this notion, defining a set to be \emph{\v{C}eby\v{s}ev relative to} another set if every point in the second set has a unique nearest neighbour'' in the first. We are interested in \v{C}eby\v{s}ev sets in some hyperspaces over $\R$, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of \v{C}eby\v{s}ev and relatively \v{C}eby\v{s}ev sets in various hyperspaces. In particular, we show that certain nested families of sets are \v{C}eby\v{s}ev. As these families are characterized purely in terms of containment, without reference to the semi-linear structure of the underlying metric space, their properties differ markedly from those of known \v{C}eby\v{s}ev sets.
 Keywords: convex body, strictly convex set, \v{C}eby\v{s}ev set, relative \v{C}eby\v{s}ev set, nested family, strongly nested family, family of translates
 MSC Classifications: 41A52 - Uniqueness of best approximation 52A20 - Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45]