Nakajima’s problem for general convex bodies

Abstract

For a convex body K C iR, the kth projection function of K assigns to any k-dimensional linear subspace of RI the k-volume of the or thogonal projection of K to that subspace. Let K and Ko be convex bodies in Rn, and let Ko be centrally symmetric and satisfy a weak regularity assump tion. Let i, j E N be such that 1 < i < j < n-2 with (i, j) :A (1, n-2). Assume that K and Ko have proportional ith projection functions and proportional jth projection functions. Then we show that K and Ko are homothetic. In the particular case where Ko is a Euclidean ball, we thus obtain characteri zations of Euclidean balls as convex bodies having constant i-brightness and constant j-brightness. This special case solves Nakajima's problem in arbitrary dimensions and for general convex bodies for most indices (i, j).