On Farthest Points of Sets
Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 163-170
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For a convex closed bounded set in a Banach space, we study the existence and uniqueness problem for a point of this set that is the farthest point from a given point in space. In terms of the existence and uniqueness of the farthest point, as well as the Lipschitzian dependence of this point on a point in space, we obtain necessary and sufficient conditions for the strong convexity of a set in several infinite-dimensional spaces, in particular, in a Hilbert space. A set representable as the intersection of closed balls of a fixed radius is called a strongly convex set. We show that the condition “for each point in space that is sufficiently far from a set, there exists a unique farthest point of the set” is a criterion for the strong convexity of a set in a finite-dimensional normed space, where the norm ball is a strongly convex set and a generating set.
Keywords:
farthest point, existence and uniqueness problem, strong convexity, Hilbert space, reflexive Banach space, proximity function.
@article{MZM_2006_80_2_a0,
author = {M. V. Balashov and G. E. Ivanov},
title = {On {Farthest} {Points} of {Sets}},
journal = {Matemati\v{c}eskie zametki},
pages = {163--170},
year = {2006},
volume = {80},
number = {2},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a0/}
}
M. V. Balashov; G. E. Ivanov. On Farthest Points of Sets. Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 163-170. http://geodesic.mathdoc.fr/item/MZM_2006_80_2_a0/