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On the generalized Busemann-Petty problem

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Abstract

The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝn with a larger volume of all i-dimensional sections necessarily have a larger volume. As proved by Bourgain and Zhang, the answer to this question is negative if i > 3. The problem is still open for i = 2, 3. In this article we prove two specific affirmative answers to the generalized Busemann-Petty problem if the body with a smaller i-dimensional volume belongs to given classes. Our results generalize Zhang’s specific affirmative answer to the generalized Busemann-Petty problem.

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Authors and Affiliations

  1. College of Mathematics and Computer Science, Chongqing Normal University, Chongqin, 400047, China

    Song-jun Lü & Gang-song Leng

  2. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Song-jun Lü

Authors
  1. Song-jun Lü
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  2. Gang-song Leng
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Correspondence to Song-jun Lü.

Additional information

This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117)

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Lü, Sj., Leng, Gs. On the generalized Busemann-Petty problem. SCI CHINA SER A 50, 1103–1116 (2007). https://doi.org/10.1007/s11425-007-0077-5

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  • DOI: https://doi.org/10.1007/s11425-007-0077-5

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