On functional stability of the solution for the problem of convex body best approximating by a ball with fixed radius

A finite-dimensional problem of finding a uniform estimate (approximation in the Hausdorff metric) of a convex body by a fixed-radius ball in an arbitrary norm is considered. It is known that this problem can be reduced to a linear programming problem in the case, when the convex body and the norm ball are polytops. Therefore, we prove the functional stability of the optimal value of the objective function with respect to accuracy of the given convex body and accuracy of the unit ball for the norm used. The stability rating is derived.