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

  • D. G. Larman (a1) and C. A. Rogers (a1)
  • DOI:
  • Published online: 01 February 2010

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.

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2.H. Busemann . “Volumes and areas of cross-sections”, Amer. Math. Monthly, 67 (1960), 248250 and 671.

3.H. Busemann . “Volumes in terms of concurrent cross-sections”, Pacific J. Math., 3 (1953), 112.

4.A. M. Davie . “The approximation problem for Banach spaces”, Bull. London Math. Soc, 5 (1973), 261266.

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