Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-17T10:26:23.306Z Has data issue: false hasContentIssue false
  • English
  • Français

NON-UNIQUENESS OF CONVEX BODIES WITH PRESCRIBED VOLUMES OF SECTIONS AND PROJECTIONS

Part of: General convexity

Published online by Cambridge University Press:  01 August 2012

Fedor Nazarov ,
Dmitry Ryabogin  and
Artem Zvavitch
Fedor Nazarov
Affiliation:
Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. (email: nazarov@math.kent.edu)
Dmitry Ryabogin
Affiliation:
Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. (email: ryabogin@math.kent.edu)
Artem Zvavitch
Affiliation:
Department of Mathematics, Kent State University, Kent, OH 44242, U.S.A. (email: zvavitch@math.kent.edu)
Get access

Abstract

We show that if d≥4 is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 52A40: Inequalities and extremum problems 52A38: Length, area, volume
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 213 - 221
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bonnesen, T. and Fenchel, W., Theory of Convex Bodies, BCS Associates (Moscow, ID, 1987).Google Scholar
[2]Gardner, R. J., Geometric Tomography, 2nd edn (Encyclopedia of Mathematics and its Applications 58), Cambridge University Press (Cambridge, 2006).Google Scholar
[3]Gardner, R. J., Ryabogin, D., Yaskin, V. and Zvavitch, A, On a problem of Klee. J. Differential Geom. (to appear) also available from http://arxiv.org/abs/1101.3364.Google Scholar
[4]Gardner, R. J. and Volčič, A., Tomography of convex and star bodies. Adv. Math. 108(2) (1994), 367–399.CrossRefGoogle Scholar
[5]Goodey, P., Schneider, R. and Weil, W., On the determination of convex bodies by projection functions. Bull. Lond. Math. Soc. 29 (1997), 82–88.Google Scholar
[6]Klee, V., Is a body spherical if its HA-measurements are constant?  Amer. Math. Monthly 76 (1969), 539–542.Google Scholar
[7]Ryabogin, D. and Yaskin, V., On counterexamples in questions of unique determination of convex bodies. Proc. Amer. Math. Soc. (to appear) also available from http://arxiv.org/abs/1201.0544.Google Scholar