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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)
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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

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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