Farthest points and strong convexity of sets
 G. E. Ivanov
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We consider the existence and uniqueness of the farthest point of a given set A in a Banach space E from a given point x in the space E. It is assumed that A is a convex, closed, and bounded set in a uniformly convex Banach space E with Fréchet differentiable norm. It is shown that, for any point x sufficiently far from the set A, the point of the set A which is farthest from x exists, is unique, and depends continuously on the point x if and only if the set A in the Minkowski sum with some other set yields a ball. Moreover, the farthest (from x) point of the set A also depends continuously on the set A in the sense of the Hausdorff metric. If the norm ball of the space E is a generating set, these conditions on the set A are equivalent to its strong convexity.
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 Title
 Farthest points and strong convexity of sets
 Journal

Mathematical Notes
Volume 87, Issue 34 , pp 355366
 Cover Date
 20100401
 DOI
 10.1134/S0001434610030065
 Print ISSN
 00014346
 Online ISSN
 15738876
 Publisher
 SP MAIK Nauka/Interperiodica
 Additional Links
 Topics
 Keywords

 optimization problem
 farthest points
 strong convexity of a set
 Banach space
 Fréchet differentiable norm
 Minkowski sum
 Hausdorff metric
 metric antiprojection
 antisun
 Industry Sectors
 Authors

 G. E. Ivanov ^{(1)}
 Author Affiliations

 1. Moscow Institute of Physics and Technology, Moscow, Russia