Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-11T02:18:25.547Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access

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

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.Ball, K. M.. Cube slicing in ℝn. Proc. Amer. Math. Soc., 97 (1986), 465–473.Google Scholar
2.Ball, K. M.. Some remarks on the geometry of convex sets. Geometric Aspects of Functional Analysis, LNM 1317 (Springer, 1988), 224–231.Google Scholar
3.Bourgain, J.. On the Busemann-Petty problem for perturbations of the ball. Geom. Fund. AnaL., 1 (1991), 1–13.CrossRefGoogle Scholar
4.Busemann, H.. Volumes in terms of concurrent cross-sections. Pacific J. Math., 3 (1953), 1–12.Google Scholar
5.Busemann, H.. Volumes and areas of cross-sections. Amer. Math. Monthly, 67 (1960), 248–250 and 671.Google Scholar
6.Busemann, H. and Petty, C. M.. Problems on convex bodies. Math. Scand., 4 (1956), 88–94.Google Scholar
7.Giannopoulos, A.. A note on a problem of Busemann, H. and Petty, C. M. concerning sections of convex symmetric bodies. Mathematika, 37 (1990), 239–244.Google Scholar
8.Larman, D. G. and Rogers, C. A.. The existence of a centrally symmetric convex body with central sections which are unexpectedly small. Mathematika, 22 (1975), 164–175.Google Scholar
9.Riesz, F. and Nagy, B. S.. Functional Analysis (Ungar, New York, 1955).Google Scholar