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

  • Michael Papadimitrakis (a1)
  • DOI:
  • Published online: 01 February 2010

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 [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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1.K. M. Ball . Cube slicing in ℝn. Proc. Amer. Math. Soc., 97 (1986), 465473.

3.J. Bourgain . On the Busemann-Petty problem for perturbations of the ball. Geom. Fund. AnaL., 1 (1991), 113.

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

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

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