Weak Concavity of the Antidistance Function

We prove that the function which for a given point of a Banach space gives the largest distance to the points of a given convex closed bounded set (antidistance) is weakly concave on the complement to some neighborhood of the set if and only if the set is a summand of some ball of some radius. We obtain precise estimates for parameters of weak concavity via the size of the neighborhood and radius of the ball in the Hilbert space.