Geodesic
Čebyšev Sets in Hyperspaces over Rn
Canadian journal of mathematics, Tome 61 (2009) no. 2, pp. 299-314

Voir la notice de l'article provenant de la source Cambridge University Press

A set in a metric space is called a Čebyšev 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 Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over ${{\text{R}}^{n}}$ , 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 Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.
DOI : 10.4153/CJM-2009-015-x
Mots-clés : 41A52, 52A20, convex body, strictly convex set, Čebyšv set, relative Čebyšv set, nested family, strongly nested family, family of translates 
Dawson, Robert J. MacG.; Moszyńska, Maria. Čebyšev Sets in Hyperspaces over Rn. Canadian journal of mathematics, Tome 61 (2009) no. 2, pp. 299-314. doi: 10.4153/CJM-2009-015-x
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