# Farthest points and strong convexity of sets

- First Online:

- Received:
- Accepted:

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.