Nakajima's Problem: Convex Bodies of Constant Width and Constant Brightness

The k th projection function of a convex body K ⊂ ℝ n 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 and curvature condition (which includes all K 0 with ∂ K 0 of class C 2 with positive radii of curvature). Assume that K and K 0 have proportional 1st projection functions ( i.e. , width functions) and proportional k th projection functions. For 2 ≤ k < ( n + 1)/2 and for k = 3, n = 5, it is shown that K and K 0 are homothetic. In the special case where K 0 is a Euclidean ball, characterizations of Euclidean balls as convex bodies of constant width and constant k ‐brightness are thus obtained.