From the Sphere to the Grassmannian: Continuous Brascamp--Lieb Inequalities via Measure Approximation and Heat Flow
Abstract
We study continuous Brascamp--Lieb inequalities associated with isotropic measures on the sphere Sn−1 and on the Grassmannian G(n,k). We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and then applies the equal-rank geometric Brascamp--Lieb inequality. In the rank-one case, the construction of the discrete isotropic measures follows from Barthe's work, which is standard, whereas in the higher-rank case the construction of finite approximate isotropic measures is given by a Tchakaloff-type theorem. Weak convergence, geometric Brascamp--Lieb inequality and Fatou's lemma then yield the continuous inequalities. The second proof generalizes the heat-semigroup preservation method of Barthe and Huet to isotropic measures on G(n,k). We prove a heat-semigroup preservation theorem in the Grassmannian setting and show that this theorem yields a Grassmannian Brascamp--Lieb inequality. At the end, we also give a sketch for the mixed-rank case of the continuous Brascamp--Lieb inequality.
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BibTeX
@misc{li2026from,
title = {{From the Sphere to the Grassmannian: Continuous Brascamp--Lieb Inequalities via Measure Approximation and Heat Flow}},
author = {Minguo Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05890},
eprint = {2610.05890},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05890},
}
RIS
TY - UNPB
TI - From the Sphere to the Grassmannian: Continuous Brascamp--Lieb Inequalities via Measure Approximation and Heat Flow
AU - Minguo Li
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05890
UR - https://arxiv.org/abs/2610.05890
AB - We study continuous Brascamp--Lieb inequalities associated with isotropic measures on the sphere S^n−1 and on the Grassmannian G(n,k). We give two proofs of the continuous Grassmannian inequality. The first proof approximates the continuous isotropic measure by discrete isotropic measures and then applies the equal-rank geometric Brascamp--Lieb inequality. In the rank-one case, the construction of the discrete isotropic measures follows from Barthe's work, which is standard, whereas in the higher-rank case the construction of finite approximate isotropic measures is given by a Tchakaloff-type theorem. Weak convergence, geometric Brascamp--Lieb inequality and Fatou's lemma then yield the continuous inequalities. The second proof generalizes the heat-semigroup preservation method of Barthe and Huet to isotropic measures on G(n,k). We prove a heat-semigroup preservation theorem in the Grassmannian setting and show that this theorem yields a Grassmannian Brascamp--Lieb inequality. At the end, we also give a sketch for the mixed-rank case of the continuous Brascamp--Lieb inequality.
ER -