Grünbaum's inequality for Gaussian and convex probability measures
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866913944083365888 |
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| author | Fradelizi, Matthieu Langharst, Dylan Liu, Jiaqian Sola, Francisco Marín Tang, Shengyu |
| author_facet | Fradelizi, Matthieu Langharst, Dylan Liu, Jiaqian Sola, Francisco Marín Tang, Shengyu |
| contents | 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 $\mathbb{R}^n$, 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-Minkowski-type 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06759 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Grünbaum's inequality for Gaussian and convex probability measures Fradelizi, Matthieu Langharst, Dylan Liu, Jiaqian Sola, Francisco Marín Tang, Shengyu Functional Analysis Metric Geometry Probability 52A40, 52A38, 46N30 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 $\mathbb{R}^n$, 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-Minkowski-type 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. |
| title | Grünbaum's inequality for Gaussian and convex probability measures |
| topic | Functional Analysis Metric Geometry Probability 52A40, 52A38, 46N30 |
| url | https://arxiv.org/abs/2507.06759 |