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

  • Fedor Nazarov (a1), Dmitry Ryabogin (a2) and Artem Zvavitch (a3)

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.

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[1]Bonnesen, T. and Fenchel, W., Theory of Convex Bodies, BCS Associates (Moscow, ID, 1987).
[2]Gardner, R. J., Geometric Tomography, 2nd edn (Encyclopedia of Mathematics and its Applications 58), Cambridge University Press (Cambridge, 2006).
[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.
[4]Gardner, R. J. and Volčič, A., Tomography of convex and star bodies. Adv. Math. 108(2) (1994), 367399.
[5]Goodey, P., Schneider, R. and Weil, W., On the determination of convex bodies by projection functions. Bull. Lond. Math. Soc. 29 (1997), 8288.
[6]Klee, V., Is a body spherical if its HA-measurements are constant?  Amer. Math. Monthly 76 (1969), 539542.
[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.
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