A Čebyšev set in a metric space is one such that every point of the space has a unique nearest neighbour in the set. In Euclidean spaces, this property is equivalent to being closed, convex, and nonempty, but in other spaces classification of Čebyšev sets may be significantly more difficult. In particular, in hyperspaces over normed linear spaces several quite different classes of Čebyšev sets are known, with no unifying description. Some new families of Čebyšev sets in hyperspaces are exhibited, with dimension d + 1 (where d is the dimension of the underlying space). They are constructed as translational closures of appropriate nested arcs
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-6,
author = {R. J. MacG. Dawson},
title = {Some \v Ceby\v sev sets with dimension d + 1 in hyperspaces over $$\mathbb{R}$^d$
},
journal = {Banach Center Publications},
volume = {86},
year = {2009},
pages = {89-110},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-6}
}
R. J. MacG. Dawson. Some Čebyšev sets with dimension d + 1 in hyperspaces over $ℝ^d$
. Banach Center Publications, Tome 86 (2009) pp. 89-110. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc84-0-6/