A STEINER TYPE FORMULA FOR CONVEX-FUNCTIONS

Given a convex function u, defined in an open bounded convex subset Om ega of R-n, we consider the set P-p(u; eta) = {x + pv: x is an element of eta, v is an element of partial derivative u(x)}, where eta is a B orel subset of Omega, p is nonnegative, and partial derivative u(x) de notes the subgradient (or subdifferential) of u at x. We prove that P- p(u; eta) is a Borel set and its n-dimensional measure is a polynomial of degree n with respect to p. The coefficients of this polynomial ar e nonnegative measures defined on the Borel subsets of Omega. We find an upper bound for the values attained by these measures on the sublev el sets of u. Such a bound depends on the quermassintegrals of the sub level set and on the Lipschitz constant of u. Finally we prove that on e of these measures coincides with the Lebesgue measure of the image u nder the subgradient map of u.