Nakajima’s Problem: Convex Bodies of Constant Width and Constant Brightness
نویسندگان
چکیده
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, 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.
منابع مشابه
Convex Bodies of Constant Width and Constant Brightness
Under the extra assumption that the boundary is of class C this was proven by S. Nakajima (= A. Matsumura) in 1926 Theorem 1 solves this problem. For convex bodies with C boundaries and positive curvature Nakajima’s result was generalized by Chakerian [?] in 1967 to “relative geometry” where the width and brightness are measured with with respect to some convex body K0 symmetric about the origi...
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