Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-23T14:40:34.359Z Has data issue: false hasContentIssue false
  • English
  • Français

The existence of a centrally symmetric convex body with central sections that are unexpectedly small

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

D. G. Larman  and
C. A. Rogers
D. G. Larman
Affiliation:
University College London
C. A. Rogers
Affiliation:
University College London
Get access

Extract

Let K, K′ be two centrally symmetric convex bodies in En, with their centres at the origin o. Let Vr denote the r-dimensional volume function. A problem of H. Busemann and C. M. Petty [1], see also, H. Busemann [2] asks:—

“If, for each (n − 1)-dimensional subspace L of En,

does it follow that

If n = 2 or, if K is an ellipsoid, then Busemann [3] shows that it does follow. However we will show that, at least for n ≥ 12, the result does not hold for general centrally symmetric convex bodies K, even if K′ is an ellipsoid. We do not construct the counter example explicitly; instead we use a probabilistic argument to establish its existence.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 22 , Issue 2 , December 1975 , pp. 164 - 175
Copyright
Copyright © University College London 1975

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.Busemann, H. and Petty, C. M.. “Problems on convex bodies”, Math. Scand., 4 (1956), 88–94.CrossRefGoogle Scholar
2.Busemann, H.. “Volumes and areas of cross-sections”, Amer. Math. Monthly, 67 (1960), 248–250 and 671.Google Scholar
3.Busemann, H.. “Volumes in terms of concurrent cross-sections”, Pacific J. Math., 3 (1953), 1–12.CrossRefGoogle Scholar
4.Davie, A. M.. “The approximation problem for Banach spaces”, Bull. London Math. Soc, 5 (1973), 261–266.CrossRefGoogle Scholar