On the generalized Busemann-Petty problem |
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Authors: | Song-jun Lü Gang-song Leng |
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Institution: | 1. College of Mathematics and Computer Science, Chongqing Normal University, Chongqin 400047, China;Department of Mathematics, Shanghai University, Shanghai 200444, China 2. College of Mathematics and Computer Science, Chongqing Normal University, Chongqin 400047, China |
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Abstract: | The generalized Busemann-Petty problem asks whether the origin-symmetric convex bodies in ℝ
n
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.
This work was supported, in part, by the National Natural Science Foundation of China (Grant No. 10671117) |
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Keywords: | star body cross i-section dual mixed volume Radon transform |
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