Farthest points and strong convexity of sets Article

First Online: 22 June 2010 Received: 05 January 2008 Accepted: 15 August 2009 DOI :
10.1134/S0001434610030065

Cite this article as: Ivanov, G.E. Math Notes (2010) 87: 355. doi:10.1134/S0001434610030065
Abstract 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.

Key words optimization problem farthest points strong convexity of a set Banach space Fréchet differentiable norm Minkowski sum Hausdorff metric metric antiprojection antisun Original Russian Text © G. E. Ivanov, 2010, published in Matematicheskie Zametki, 2010, Vol. 87, No. 3, pp. 382–395.

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Authors and Affiliations 1. Moscow Institute of Physics and Technology Moscow Russia