Mathematika

Research Article

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

D. G. Larmana1 and C. A. Rogersa1

a1 University College London

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,

S0025579300006033_eqnU1

does it follow that

S0025579300006033_eqnU2

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.

(Received August 19 1975)

Key Words:

  • 52A20: CONVEX SETS AND GEOMETRIC INEQUALITIES; Convex sets in n-dimensions.