论文标题
busemann-petty问题在日志concave功能的熵上
The Busemann-Petty problem on entropy of log-concave functions
论文作者
论文摘要
Busemann-Petty问题询问对称凸面是否在 Euclidean Space $ \ Mathbb {r}^n $带有较小的中央超平面部分必须具有较小的体积。解决方案已经完成，如果$ n \ le 4 $且负数为$ n \ ge 5 $，则答案是肯定的。在本文中，我们在log-concave函数的熵上调查了Busemann-Petty问题：对于log-concave函数$ f $和$ g $，在$ \ mathbb {r}^n $中具有有限的正积分，如果marginal $ \ int $ \ int _ { $ \ int _ {\ mathbb {r}^n \ cap h} g（x）dx $ $ g $ $ g $的每个超平面$ h $，无论是熵$ {\ rm ent} $ $ f $的$ {\ rm ent}（f $ of $ f $的$ f $都比entropy $ {\ rm ent} $ {\ rm ent}（g $ g $ g $ g $ g $ g $ g $吗？ busemann-petty问题在日志concave功能的熵上包括Busemann-Petty问题，因此，当$ n \ geq5 $时，其答案为负。对于$ 2 \ leq n \ leq4 $，我们在log-concove函数的熵上为Busemann-Petty问题提供了积极的答案。
The Busemann-Petty problem asks whether symmetric convex bodies in the Euclidean space $\mathbb{R}^n$ with smaller central hyperplane sections necessarily have smaller volume. The solution has been completed and the answer is affirmative if $n \le 4$ and negative if $n\ge 5$. 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 $\mathbb{R}^n$, 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, whether the entropy ${\rm Ent}(f)$ of $f$ is bigger than the entropy ${\rm Ent}(g)$ of $g$? The Busemann-Petty problem on entropy of log-concave functions includes the Busemann-Petty problem, hence, its answer is negative when $n\geq5$. For $2\leq n\leq4$ we give a positive answer to the Busemann-Petty problem on entropy of log-concave functions.
