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A problem of klee on inner section functions of convex bodies

Richard J. Gardner Western Washington University Dmitry Ryabogin Kent State University Vlad Yaskin University of Alberta Artem Zvavitch Kent State University

Differential Geometry mathscidoc:1609.10271

Journal of Differential Geometry, 91, (2), 261-279, 2012
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 xn-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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@inproceedings{richard2012a,
  title={A PROBLEM OF KLEE ON INNER SECTION FUNCTIONS OF CONVEX BODIES},
  author={Richard J. Gardner, Dmitry Ryabogin, Vlad Yaskin, and Artem Zvavitch},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913230315764846933},
  booktitle={Journal of Differential Geometry},
  volume={91},
  number={2},
  pages={261-279},
  year={2012},
}
Richard J. Gardner, Dmitry Ryabogin, Vlad Yaskin, and Artem Zvavitch. A PROBLEM OF KLEE ON INNER SECTION FUNCTIONS OF CONVEX BODIES. 2012. Vol. 91. In Journal of Differential Geometry. pp.261-279. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160913230315764846933.
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