Farthest points and strong convexity of sets
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- Ivanov, G.E. Math Notes (2010) 87: 355. doi:10.1134/S0001434610030065
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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.