Hessian Valuations

A new class of continuous valuations on the space of convex functions on R-n is introduced. On smooth convex functions, they are defined for i = 0,..., n byu -> integral(Rn)zeta(u(x),x, del u(x))[D(2)u(x)](i)dxwhere zeta is an element of C(R x R-n x R-n) and [D(2)u](i) is the ith elementary symmetric function of the eigenvalues of the Hessian matrix, D(2)u, of u. Under suitable assumptions on zeta, these valuations are shown to be invariant under translations and rotations on convex and coercive functions.