Grünbaum's inequality for Gaussian and convex probability measures
Abstract
A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities -with equality characterizations -for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an "Ehrhard-Grünbaum" inequality for the Gaussian measure on Rn , which improves upon the bound derived from its log-concavity. For s-concave Radon measures, our framework provides a simpler proof of known results and, more importantly, yields the previously missing equality characterization. This is achieved by gaining new insight into the equality case of their Brunn-Minkowskitype inequality. Moreover, we show that these "s-Grünbaum" inequalities can hold only when s > -1. However, for convex measures on the real line, we prove Grünbaum-type inequalities involving their cumulative distribution function.
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BibTeX
@misc{fradelizi2026grünbaums,
title = {{Grünbaum's inequality for Gaussian and convex probability measures}},
author = {Matthieu Fradelizi and Dylan Langharst and Jiaqian Liu and Francisco Marín Sola and Shengyu Tang},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.1016/j.matpur.2026.103938},
eprint = {2507.06759},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05777267v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Grünbaum's inequality for Gaussian and convex probability measures
AU - Matthieu Fradelizi
AU - Dylan Langharst
AU - Jiaqian Liu
AU - Francisco Marín Sola
AU - Shengyu Tang
PY - 2026
DA - 2026/10/04
PB - HAL
DO - 10.1016/j.matpur.2026.103938
UR - https://hal.science/hal-05777267v1
AB - A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities -with equality characterizations -for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an "Ehrhard-Grünbaum" inequality for the Gaussian measure on R^n , which improves upon the bound derived from its log-concavity. For s-concave Radon measures, our framework provides a simpler proof of known results and, more importantly, yields the previously missing equality characterization. This is achieved by gaining new insight into the equality case of their Brunn-Minkowskitype inequality. Moreover, we show that these "s-Grünbaum" inequalities can hold only when s > -1. However, for convex measures on the real line, we prove Grünbaum-type inequalities involving their cumulative distribution function.
ER -