Nakajima’s Problem for General Convex Bodies

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, let K0 be centrally symmetric and satisfy a weak regularity assumption. Let i, j ∈ N be such that 1 ≤ i < j ≤ n − 2 with (i, j) 6= (1, n−2). Assume that K and K0 have proportional ith projection functions and proportional jth projection functions. Then we show thatK andK0 are homothetic. In the particular case where K0 is a Euclidean ball, we thus obtain characterizations 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).