AI 中文总结
研究对数凹函数情形下张投影不等式的函数形式，通过考虑离散测度证明其离散版本，包括得到贝尔瓦尔德不等式和罗杰斯 - 谢泼德不等式的离散形式，且离散版本蕴含连续形式。
AI 中文摘要
本文关注一个对任意具有正积分的可积对数凹函数\(f\)成立的不等式，它给出了\(f\)的协方差图函数的第\(n\)个球状体与其极投影体之间的包含关系，在对数凹函数情形下提供了张投影不等式的函数形式。我们将证明该不等式的离散版本，考虑涉及格点枚举测度的离散测度而非勒贝格测度。为此，会得到贝尔瓦尔德不等式函数形式的离散版本，还给出罗杰斯 - 谢泼德不等式的离散版本。所有不等式的离散版本都蕴含其连续形式。
英文摘要
In this paper, we focus our attention on the following inequality, which holds for any integrable log-concave function with positive integral $f$, and gives an inclusion between the $n$-th Ball body of the covariogram function of $f$ and its polar projection body, providing a functional version of Zhang's projection inequality in the setting of log-concave functions: $$ n\int_0^\infty r^{n-1}\int_{\mathbb{R}^n}\min\{f(z),f(z-re_n)\}dzdr\leq n!\frac{\left(\int_{\mathbb{R}^n}f(x)dx\right)^{n+1}}{\left(\int_{e_n^\perp}P_{e_n^\perp}f(y)dy\right)^n}. $$ Here $(e_i)_{i=1}^n$ denotes the canonical basis in $\mathbb{R}^n$ and $P_{e_n^\perp} f$ denotes the projection of $f$ onto the hyperplane orthogonal to $e_n$ given, for $y\in e_n^\perp$, by $P_{e_n^\perp} f(y)=\sup_{λ\in\mathbb{R}}f(y+λe_n)$. We will prove a discrete version of this inequality in which we will consider discrete measures involving the lattice point enumerator measure, rather than the Lebesgue measure. In order to prove it, we will obtain a discrete version of the functional version of Berwald's inequality. We will also provide a discrete version of the functional Rogers-Shephard inequality. All the discrete versions of the inequalities considered will imply their continuous counterparts.
Comments35 pages
