Article

Mathematical Notes

, Volume 87, Issue 3, pp 355-366

Farthest points and strong convexity of sets

  • G. E. IvanovAffiliated withMoscow Institute of Physics and Technology Email author 

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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