Let A and B be two compact, convex sets in ℝn, each symmetric with respect to the origin 0. L is any (n - l)-dimensional subspace. In 1956 H. Busemann and C. M. Petty (see ) raised the question: Does vol (A ⌒ L) < vol (B ⌒ L) 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 ) proved that if A is any ellipsoid the answer is affirmative. In fact, as he observed in , the answer is still affirmative if A is an ellipsoid with 0 as center of symmetry and B is any compact set containing 0.
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