  We study the Hausdorff distance between a set and its convex hull. Let $X$ be
a Banach space, define the CHD-module of space $X$ as the supremum of this
distance for all subset of the unit ball in $X$. In the case of finite
dimensional Banach spaces we obtain the exact upper bound of the CHD-module
depending on the dimension of the space. We give an upper bound for the
CHD-module in $L_p$ spaces. We prove that CHD-module is not greater than the
maximum of the Lipschitz constants of metric projection operator onto
hyperplanes. This implies that for a Hilbert space CHD-module equals 1. We
prove criterion of the Hilbert space and study the contractibility of
proximally smooth sets in uniformly convex and uniformly smooth Banach spaces.
