Nakajima's problem: convex bodies of constant width and constant brightness

Original · EN

For a convex body K⊂ⁿ, the kth projection function of K assigns to any k-dimensional linear subspace of ⁿ the k-volume of the orthogonal projection of K to that subspace. Let K and K₀ be convex bodies in ⁿ, and let K₀ be centrally symmetric and satisfy a weak regularity and curvature condition (which includes all K₀ with K₀ of class C² with positive radii of curvature). Assume that K and K₀ have proportional 1st projection functions (i.e., width functions) and proportional kth projection functions. For 2≤ k<(n+1)/2 and for k=3, n=5 we show that K and K₀ are homothetic. In the special case where K₀ is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies of constant width and constant k-brightness.