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1. CJM 2009 (vol 61 pp. 299)

Dawson, Robert J. MacG.; Moszy\'{n}ska, Maria
\v{C}eby\v{s}ev Sets in Hyperspaces over $\mathrm{R}^n$
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
Categories:41A52, 52A20

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