Linear Wavefronts of Convex Polyhedra

Let M ⊂ ℝn be a convex polyhedron, i.e., the intersection of a finite number of closed half-spaces that is bounded and has nonempty interior. Let each hyperplane of the hyperfaces f1,.. , fm of M move inwards M in a self-parallel fashion at a constant nonnegative speed (it is assumed that at least one face has nonzero speed). This yields a “shrinking” polyhedron. Let reg(f1),.. , reg(fm) be the parts of M (with disjoint interiors) swept by the faces f1,.. , fm during the “shrinking” process. The main result is as follows. Let F be a functional on the class of convex compact subsets in ℝn. It is assumed 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 each m-tuple (x1,.. , xm) of nonnegative reals with nonzero sum there exists an m-tuple of “velocities” for the faces f1,.. , fm such that the m-tuple (F(reg(f1)),.. , F(reg(fm))) is proportional to (x1,.. , xm). Bibliography: 1 title. © 2016, Springer Science+Business Media New York.