A ball of maximal radius inscribed in a convex closed bounded set with a nonempty interior is considered in the class of uniformly convex Banach spaces. It is shown that, under certain conditions, the centers of inscribed balls form a uniformly continuous (as a set function) set-valued mapping in the Hausdorff metric. In a finite-dimensional space of dimension n, the set of centers of balls inscribed in polyhedra with a fixed collection of normals satisfies the Lipschitz condition with respect to sets in the Hausdorff metric. A Lipschitz continuous single-valued selector of the set of centers of balls inscribed in such polyhedra can be found by solving linear programming problems.