On linear wavefronts of convex polyhedra

By a convex polyhedron we mean the intersection of a finite number of closed half-spaces in a Euclidean space whenever this intersection is bounded and has non-empty interior.
Let each hyperplane of the hyperfaces $f_1,\dots,f_m$ of a polyhedron $M$ in $\mathbb R^n$ move inwards $M$ in a self-parallel fashion at a non-negative constant speed (we assume that at least one face has non-zero speed). We thus obtain a “shrinking” polyhedron. Let $\operatorname{reg}(f_1),\dots,\operatorname{reg}(f_m)$ be the parts of $M$ (with disjoint interiors) that the faces $f_1,\dots,f_m$ sweep during the “shrinking” process.
The main result is as follows. Let $F$ be a functional on the class of convex compact subsets in $\mathbb R^n$. We assume that $F$ is nonnegative and continuous (with respect to the Hausdorff metric), and, furthermore, $F(K)=0$ if and only if $\dim(K)<n$. Then for every $m$-tuple $(x_1,\dots,x_m)$ of nonnegative reals with non-zero sum there exists an $m$-tuple of “speeds” for the faces $f_1,\dots,f_m$ such that the $m$-tuple $(F(\operatorname{reg}(f_1)),\dots,F(\operatorname{reg}(f_m)))$ is proportional to $(x_1,\dots,x_m)$.