A problem of Klee on inner section functions of convex bodies

2011/01/18 by Richard J. Gardner, Gardner, Richard J., Dmitri Ryabogin +5
Biochemistry, Genetics and Molecular Biology · Mathematics · #52A20 #52A38 #52A40 #Classical Analysis and ODEs (math.CA) #Diffusion and Search Dynamics #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1101.3364

openalex publication_date 2011/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1969, Vic Klee asked whether a convex body is uniquely determined (up to translation and reflection in the origin) by its inner section function, the function giving for each direction the maximal area of sections of the body by hyperplanes orthogonal to that direction. We answer this question in the negative by constructing two infinitely smooth convex bodies of revolution about the x_n-axis in \Rⁿ, n≥ 3, one origin symmetric and the other not centrally symmetric, with the same inner section function. Moreover, the pair of bodies can be arbitrarily close to the unit ball.

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