For a convex body K ⊂ Rn and i ∈ {1, . . . , n− 1}, the function assigning to any i-dimensional subspace L of Rn, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K0 ⊂ Rn be smooth convex bodies of class C2 +, and let K0 be centrally symmetric. Excluding two exceptional cases, we prove that K and K0 are homothetic if they have ...

Goodey and Weil have recently introduced the notions of translation mixtures of convex bodies and of mixed convex bodies. By a new approach, a simpler proof for the existence of the mixed polytopes is given, and explicit formulae for their vertices and edges are obtained. Moreover, the theory of mixed bodies is extended to more than two convex bodies. The paper concludes with the proof of an in...

For a convex body K ⊂ Rn, the kth projection function of K assigns to any k-dimensional linear subspace of Rn the k-volume of the orthogonal projection of K to that subspace. Let K and K0 be convex bodies in Rn, and let K0 be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K0 with ∂K0 of class C2 with positive radii of curvature). Assume that K and ...