Nakajima's problem for general convex bodies

Daniel Hug
2008 Proceedings of the American Mathematical Society  
For a convex body K ⊂ R n , the kth projection function of K assigns to any k-dimensional linear subspace of R n the k-volume of the orthogonal projection of K to that subspace. Let K and K 0 be convex bodies in R n , and let K 0 be centrally symmetric and satisfy a weak regularity assumption. Let i, j ∈ N be such that 1 ≤ i < j ≤ n−2 with (i, j) = (1, n−2). Assume that K and K 0 have proportional ith projection functions and proportional jth projection functions. Then we show that K and K 0
more » ... w that K and K 0 are homothetic. In the particular case where K 0 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).
doi:10.1090/s0002-9939-08-09432-x fatcat:hybjvyztjvax7lqew2linrbmom