Geodesic
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. 
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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/

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