Linear Wavefronts of Convex Polyhedra

Abstract

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, . . . , f_m 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(f_m) 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, . . . , x_m) of nonnegative reals with nonzero sum there exists an m-tuple of “velocities” for the faces f₁, . . . , f_m such that the m-tuple (F(reg(f₁)), . . . , F(reg(fm))) is proportional to (x₁, . . . , x_m). Bibliography: 1 title.