On cost-induced Santaló-type inequalities in Polish measure spaces
Abstract
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 RCD(K,∞) 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.
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BibTeX
@misc{langharst2026costinduced,
title = {{On cost-induced Santaló-type inequalities in Polish measure spaces}},
author = {Dylan Langharst and Andreas Malliaris and Michael Roysdon},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01460},
eprint = {2609.01460},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01460},
}
RIS
TY - UNPB
TI - On cost-induced Santaló-type inequalities in Polish measure spaces
AU - Dylan Langharst
AU - Andreas Malliaris
AU - Michael Roysdon
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.01460
UR - https://arxiv.org/abs/2609.01460
AB - 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 RCD(K,∞) 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.
ER -