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On the Busemann-Petty problem about convex, centrally symmetric bodies in ℝn

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Michael Papadimitrakis
Michael Papadimitrakis
Affiliation:
Department of Mathematics, Washington University, Campus Box 1146, One Brookings Drive, St. Louis, MO 63130-4899, U.S.A.
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Extract

Let A and B be two compact, convex sets in ℝn, each symmetric with respect to the origin 0. L is any (n - l)-dimensional subspace. In 1956 H. Busemann and C. M. Petty (see [6]) raised the question: Does vol (A ⌒ L) < vol (B ⌒ L) for every L imply vol (A) < vol(B)? The answer in case n = 2 is affirmative in a trivial way. Also in 1953 H. Busemann (see [4]) proved that if A is any ellipsoid the answer is affirmative. In fact, as he observed in [5], the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 39 , Issue 2 , December 1992 , pp. 258 - 266
Copyright
Copyright © University College London 1992

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References

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