On Farthest Points of Sets
Matematičeskie zametki, Tome 80 (2006) no. 2, pp. 163-170
Voir la notice de l'article provenant de la source Math-Net.Ru
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},
publisher = {mathdoc},
volume = {80},
number = {2},
year = {2006},
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/