Deviation of the convex hull of bounded sets

We study the maximal sets convex hull deviation (CHD) from the set provided that the set is contained in the unit ball. For the finite-dimensional space we obtain the exact upper bound of the CHD depending on the dimension of the space. We give an upper bound of the CHD via the Lipschitz constant of the metric projection operator onto the hyperplane. This Lipschitz constant, in turns, is bounded from above via the modules of smoothness and convexity of the space.