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A logarithmic Brascamp--Lieb variance inequality
Abstract
We prove a strengthened Brascamp--Lieb inequality for even log-concave measures whose positive sublevel bodies satisfy the local logarithmic Brunn--Minkowski inequality. The conclusion can be written either by adding a rank-one term to the Hessian of the potential or by adding an explicit nonnegative remainder to the variance. The proof uses sublevel volume variations, a one-dimensional Brascamp--Lieb inequality, and the coarea formula. The derivative of the potential under dilations gives an equality function whenever it is square-integrable. For potentials homogeneous of degree greater than one, we obtain a sharp radial remainder.
Related
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BibTeX
@misc{kolesnikov2026logarithmic,
title = {{A logarithmic Brascamp--Lieb variance inequality}},
author = {Alexander V. Kolesnikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21669},
eprint = {2610.21669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21669},
}
RIS
TY - UNPB
TI - A logarithmic Brascamp--Lieb variance inequality
AU - Alexander V. Kolesnikov
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21669
UR - https://arxiv.org/abs/2610.21669
AB - We prove a strengthened Brascamp--Lieb inequality for even log-concave measures whose positive sublevel bodies satisfy the local logarithmic Brunn--Minkowski inequality. The conclusion can be written either by adding a rank-one term to the Hessian of the potential or by adding an explicit nonnegative remainder to the variance. The proof uses sublevel volume variations, a one-dimensional Brascamp--Lieb inequality, and the coarea formula. The derivative of the potential under dilations gives an equality function whenever it is square-integrable. For potentials homogeneous of degree greater than one, we obtain a sharp radial remainder.
ER -