Farthest Points and Strong Convexity of Sets

We consider the existence and uniqueness of the farthest point of a given set $A$ in 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.