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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
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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

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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