Smooth convex bodies with proportional projection functions

Abstract

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 with boundaries of class C2 and positive Gauss-Kronecker curvature and assume K0 is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K0 are homothetic if their i-th and j-th projection functions are proportional. When K0 is a Euclidean ball this shows that a convex body with C2 boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball.