发表机构
Carnegie Mellon University; University of Toulouse & CNRS; University of Cincinnati; The Ohio State University（卡内基梅隆大学; 图卢兹大学及法国国家科学研究中心; 辛辛那提大学; 俄亥俄州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了成本-Santaló不等式框架，通过迁移原理推导新的泛函Santaló不等式，应用于RCD(K,∞)空间、汉明立方体及最优传输领域，拓展了相关理论并得到具体实例。
AI 中文摘要
我们引入了一个框架，用于在由连续成本函数耦合在一起的波兰测度空间的m元组上建立Blaschke-Santaló型不等式。我们方法的核心是一个迁移原理，它提供了一种机制，将涉及成本极集的几何加权不等式提升为Santaló型的泛函积分不等式。我们将这些等价不等式称为成本-Santaló不等式。这一定义扩展并包含了文献中先前的概念。我们应用该原理推导出几个新的泛函Santaló不等式版本，包括在矩形矩阵空间和泛函正弦Santaló不等式上的版本。一个令人惊讶的进展是，具有对数凹等周函数的概率空间，例如高斯空间和球面空间，可纳入我们的框架，从而在这些场景中得到新的泛函Santaló不等式。特别地，我们获得了关于RCD(K,∞)空间的结果。作为一个离散应用，我们得到了关于汉明立方体的一个不等式。最后，我们探索了在最优传输中的应用，利用我们的泛函框架在满足成本-Santaló不等式的任意波兰空间上建立广义传输-熵不等式，我们在矩阵空间上明确实例化了这一点。
英文摘要
We introduce a framework for establishing Blaschke-Santaló-type inequalities on $m$-tuples of Polish measure spaces coupled together by a continuous cost function. Central to our approach is a transference principle, which provides a mechanism to lift geometric weighted inequalities involving cost-polar sets into functional integral inequalities of Santaló-type. We call these equivalent inequalities cost-Santaló inequalities. This definition expands and includes previous notions in the literature. We apply this principle to deduce several new versions of functional Santaló inequalities, including on the space of rectangular matrices and a functional sine Santaló inequality. A surprising development is that probability spaces with log-concave isoperimetric functions fit into our framework, for example, Gauss space and spherical space, leading to new functional Santaló inequalities in these settings. In particular, we obtain results for $\operatorname{RCD}(K,\infty)$ spaces. As a discrete application, we obtain an inequality for the Hamming cube. Finally, we explore applications to optimal transport, utilizing our functional framework to establish generalized transport-entropy inequalities on arbitrary Polish spaces satisfying a cost-Santaló inequality, which we explicitly instantiate for matrix spaces.
Comments62 pages; v2: A.M. added his funding
