Mathematika

Research Article

On the Busemann-Petty problem about convex, centrally symmetric bodies in n

Dr Michael Papadimitrakisa1

a1 Department of Mathematics, Washington University, Campus Box 1146, One Brookings Drive, St. Louis, MO 63130-4899, U.S.A.

Let A and B be two compact, convex sets in n, 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 (AL) < vol (BL) 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.

(Received November 08 1991)

Key Words:

  • 52A20: CONVEX AND DISCRETE GEOMETRY; General convexity; Convex sets in n-dimensions.