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

2006/01/20 by Ralph Howard, Howard, Ralph, Daniel Hug +1
Biochemistry, Genetics and Molecular Biology · Mathematics · #52A20 #Diffusion and Search Dynamics #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.math/0601499

openalex publication_date 2006/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 \f 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

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