Extremal Problems for the Distance Between Elements of Two Sets

Given two sets A and C in a Banach space, we consider four extremal problems for the distance between two elements, one from A and the other from C. The first problem is to minimize the distance by choosing elements from these sets; the second problem is to maximize it; the third one is the minimax problem; the fourth one is the maximin problem. These problems arise in approximation theory and constrained optimization, they are generalizations of the best approximation problem and the problem of farthest points. In terms of prox-regularity and of the property of being a summand of the ball for sets A and C, we obtain sharp sufficient conditions for each of the problems to have a unique solution and, moreover, to be Tykhonov well-posed. We also develop the calculus of convexity parameters for subsets of a Banach space in connection with the Minkowski sum and difference.