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An asymmetric convex body with maximal sections of constant volume


Authors: Fedor Nazarov, Dmitry Ryabogin and Artem Zvavitch
Journal: J. Amer. Math. Soc. 27 (2014), 43-68
MSC (2010): Primary 52A20, 52A40; Secondary 52A38
DOI: https://doi.org/10.1090/S0894-0347-2013-00777-8
Published electronically: July 18, 2013
MathSciNet review: 3110795
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Abstract: We show that in all dimensions $ d\ge 3$, there exists an asymmetric convex body of revolution all of whose maximal hyperplane sections have the same volume. This gives the negative answer to the question posed by V. Klee in 1969.


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Additional Information

Fedor Nazarov
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: nazarov@math.kent.edu

Dmitry Ryabogin
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: ryabogin@math.kent.edu

Artem Zvavitch
Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
Email: zvavitch@math.kent.edu

DOI: https://doi.org/10.1090/S0894-0347-2013-00777-8
Keywords: Convex bodies, sections, projections
Received by editor(s): January 6, 2012
Received by editor(s) in revised form: October 4, 2012, and May 31, 2013
Published electronically: July 18, 2013
Additional Notes: The first author is supported in part by U.S. National Science Foundation Grant DMS-0800243
The second and third authors are supported in part by U.S. National Science Foundation Grant DMS-1101636.
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.