An asymmetric convex body with maximal sections of constant volume
HTML articles powered by AMS MathViewer
- by Fedor Nazarov, Dmitry Ryabogin and Artem Zvavitch
- J. Amer. Math. Soc. 27 (2014), 43-68
- DOI: https://doi.org/10.1090/S0894-0347-2013-00777-8
- Published electronically: July 18, 2013
- PDF | Request permission
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.References
- S. Banach, $http://en.wikipedia.org/wiki/Banach\_fixed$-$point\_theorem$.
- T. Bonnesen and Fenchel, Theory of convex bodies, Moscow, Idaho, 1987.
- Richard J. Gardner, Geometric tomography, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, New York, 2006. MR 2251886, DOI 10.1017/CBO9781107341029
- Richard J. Gardner, Dmitry Ryabogin, Vlad Yaskin, and Artem Zvavitch, A problem of Klee on inner section functions of convex bodies, J. Differential Geom. 91 (2012), no. 2, 261–279. MR 2971289
- Paul Goodey, Rolf Schneider, and Wolfgang Weil, On the determination of convex bodies by projection functions, Bull. London Math. Soc. 29 (1997), no. 1, 82–88. MR 1416411, DOI 10.1112/S0024609396001968
- Sigurdur Helgason, The Radon transform, 2nd ed., Progress in Mathematics, vol. 5, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1723736, DOI 10.1007/978-1-4757-1463-0
- Victor Klee, Research Problems: Is a Body Spherical If Its $HA$-Measurements are Constant?, Amer. Math. Monthly 76 (1969), no. 5, 539–542. MR 1535423, DOI 10.2307/2316970
- Jiří Matoušek, Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry; Written in cooperation with Anders Björner and Günter M. Ziegler. MR 1988723
- Fedor Nazarov, Dmitry Ryabogin, and Artem Zvavitch, Non-uniqueness of convex bodies with prescribed volumes of sections and projections, Mathematika 59 (2013), no. 1, 213–221. MR 3028180, DOI 10.1112/S0025579312001027
- Dmitry Ryabogin and Vlad Yaskin, On counterexamples in questions of unique determination of convex bodies, Proc. Amer. Math. Soc. 141 (2013), no. 8, 2869–2874. MR 3056577, DOI 10.1090/S0002-9939-2013-11539-X
Bibliographic Information
- Fedor Nazarov
- Affiliation: Department of Mathematics, Kent State University, Kent, Ohio 44242
- MR Author ID: 233855
- 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
- MR Author ID: 671170
- Email: zvavitch@math.kent.edu
- 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. - © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3110795