格伦鲍姆不等式的熵类似物
Entropic analogues of Grünbaum's inequality
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中文总结 AI 辅助
该研究受格伦鲍姆不等式启发，针对对数凹随机变量X，证明了关于熵的类似不等式，包括h(X) - (e/(e - 1))H₂(1/e) ≤ h(X|X ≤ EX) ≤ h(X)，推广了雷尼熵上界和最小熵下界，刻画等号成立情况，还讨论了高维推广及反例。
中文摘要 AI 辅助
经典的格伦鲍姆不等式表明，被包含其重心的半空间截断的凸体体积比例至少为1/e。从其函数对应物可知，对于任何对数凹随机变量X，有P(X≥EX)≥1/e，当且仅当X为指数分布时取等号。受凸体格伦鲍姆不等式及其函数推广的启发，我们证明了熵的类似不等式，并刻画了等号成立的情况。我们表明，如果X是R上的对数凹随机变量，那么h(X) - (e/(e - 1))H₂(1/e) ≤ h(X|X ≤ EX) ≤ h(X)，其中h是微分熵，H₂(·)是二元熵函数，X|X≤EX表示X在X≤EX条件下的分布。我们推广了所有雷尼熵的上界和最小熵的下界，并刻画了所有等号成立的情况，还讨论了高维中的潜在推广及一些方向上的反例。作为证明下界的中间步骤，我们建立了一个新不等式，使用自由度技术结合标准KKT型优化引理进行证明，同时刻画了微分熵和最小熵之间已知比较不等式中等号成立的情况。
英文摘要
The classical Grünbaum inequality asserts that the proportion of the volume of a convex body cut off by a halfspace containing its barycenter is at least $1/e$. From its functional counterpart, for any log-concave random variable $X$, one has $\mathbb{P}(X\ge \mathbb{E}X)\ge 1/e$, with equality if and only if $X$ is exponential. Motivated by Grünbaum's inequality for convex bodies and its functional generalizations, we prove analogous inequalities for entropy, with characterizations of the equality cases. We show that if $X$ is a log-concave random variable on $\mathbb{R}$, then $$ h(X)-\frac{e}{e-1}H_2(1/e) \leq h(X|X \leq \mathbb{E}X) \leq h(X), $$ where $h$ is the differential entropy, $H_2(\cdot)$ is the binary entropy function and $X|X\leq \mathbb{E}X$ stands for the distribution of $X$ conditional on $X\leq \mathbb{E}X$. We generalize the upper bound for all Rényi entropies and the lower bound for min-entropy. Our inequalities are sharp and we characterize all equality cases. We discuss potential generalizations in high dimensions and give counterexamples in some directions. As an intermediate step for the proof of the lower bound, we establish a new inequality that we prove using a technique known as degrees of freedom, combined with a standard KKT-type optimization lemma. Along the way, we characterize the equality case in a known comparison inequality between differential and min-entropy, which may be of independent interest.
