Open Access
June 2012 A problem of Klee on inner section functions of convex bodies
Richard J. Gardner, Dmitry Ryabogin, Vlad Yaskin, Artem Zvavitch
J. Differential Geom. 91(2): 261-279 (June 2012). DOI: 10.4310/jdg/1344430824


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 con- structing two infinitely smooth convex bodies of revolution about the $x_n$-axis in $\mathbb{R}^n, n\ge 3$, 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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Richard J. Gardner. Dmitry Ryabogin. Vlad Yaskin. Artem Zvavitch. "A problem of Klee on inner section functions of convex bodies." J. Differential Geom. 91 (2) 261 - 279, June 2012.


Published: June 2012
First available in Project Euclid: 8 August 2012

zbMATH: 1255.52005
MathSciNet: MR2971289
Digital Object Identifier: 10.4310/jdg/1344430824

Rights: Copyright © 2012 Lehigh University

Vol.91 • No. 2 • June 2012
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