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The Busemann-Petty problem on entropy of log-concave functions

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Abstract

The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space ℝnwith smaller central hyperplane sections necessarily have smaller volumes. The solution has been completed and the answer is affirmative if n ⩽ 4 and negative if n ⩾ 5. In this paper, we investigate the Busemann-Petty problem on entropy of log-concave functions: for even log-concave functions f and g with finite positive integrals in ℝn, if the marginal \(\int_{{\mathbb{R}^n} \cap H} {f(x)dx} \) of f is smaller than the marginal \(\int_{{\mathbb{R}^n} \cap H} {g(x)dx} \) of g for every hyperplane H passing through the origin, is the entropy Ent(f) of f bigger than the entropy Ent(g) of g? The Busemann-Petty problem on entropy of log-concave functions includes the Busemann-Petty problem, and hence its answer is negative when n ⩾ 5. For 2 ⩽ n ⩽ 4, we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.

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Acknowledgements

The first author was supported by National Natural Science Foundation of China (Grant No. 12001291) and the Fundamental Research Funds for the Central Universities (Grant No. 531118010593). The second author was supported by National Natural Science Foundation of China (Grant No. 12071318). The authors thank anonymous referees for helpful suggestions and comments that directly led to the improvement of the early manuscript.

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Authors and Affiliations

  1. School of Mathematics, Hunan University, Changsha, 410082, China

    Niufa Fang

  2. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

    Jiazu Zhou

  3. College of Science, Wuhan University of Science and Technology, Wuhan, 430081, China

    Jiazu Zhou

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  1. Niufa Fang
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Correspondence to Jiazu Zhou.

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Fang, N., Zhou, J. The Busemann-Petty problem on entropy of log-concave functions. Sci. China Math. 65, 2171–2182 (2022). https://doi.org/10.1007/s11425-021-1907-6

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