A representation for the support function of a convex body

In this paper a formula for a translation invariant measure of planes intersecting a $n$-dimensional convex body in terms of curvatures of $2$-dimensional projections of the body was found. The paper also gives a new simple proof of the representation for the support function of an origin symmetric $3$-dimensional convex body, which was obtained by means of a stochastic approximation of the convex body.