Approximation of continuous set-valued maps by constant set-valued maps with image balls

It is shown that the problem of the best uniform approximatiou in the Hausdorff metric Of a continous set-valued map with finite-dimensional compact convex images by constant set-valued maps whose images are balls in some norm can be reduced to a Visual geometric problem. The latter consists in constructing a spherical layer of minimal thickness which contains the complement of a compact convex set to a larger compact convex set.