Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866914411919179776 |
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| author | Aishwarya, Gautam Li, Dongbin |
| author_facet | Aishwarya, Gautam Li, Dongbin |
| contents | Given even strongly log-concave random vectors $X_{0}$ and $X_{1}$ in $\mathbb{R}^n$, we show that a natural joint distribution $(X_{0},X_{1})$ satisfies, \begin{equation}
e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \geq (1-t) e^{ - \frac{1}{n}D (X_{0} \Vert Z)} + t e^{ - \frac{1}{n}D ( X_{1} \Vert Z)}, \end{equation} where $Z$ is distributed according to the standard Gaussian measure $γ$ on $\mathbb{R}^n$, $t \in [0,1]$, and $D(\cdot \Vert Z)$ is the Gaussian relative entropy. This extends and provides a different viewpoint on the corresponding geometric inequality proved by Eskenazis and Moschidis, namely that \begin{equation} γ\left( (1-t) K_{0} + t K_{1} \right)^{\frac{1}{n}} \geq (1-t) γ(K_{0})^{\frac{1}{n}} + t γ(K_{1})^{\frac{1}{n}}, \end{equation} when $K_{0}, K_{1} \subseteq \mathbb{R}^n$ are origin-symmetric convex bodies. As an application, using Donsker--Varadhan duality, we obtain Gaussian Borell--Brascamp--Lieb inequalities applicable to even log-concave functions, which serve as functional forms of the Eskenazis--Moschidis inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_03114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space Aishwarya, Gautam Li, Dongbin Probability Functional Analysis Metric Geometry 37C10, 94A17, 52A40, 52A20, 49Q22 Given even strongly log-concave random vectors $X_{0}$ and $X_{1}$ in $\mathbb{R}^n$, we show that a natural joint distribution $(X_{0},X_{1})$ satisfies, \begin{equation} e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \geq (1-t) e^{ - \frac{1}{n}D (X_{0} \Vert Z)} + t e^{ - \frac{1}{n}D ( X_{1} \Vert Z)}, \end{equation} where $Z$ is distributed according to the standard Gaussian measure $γ$ on $\mathbb{R}^n$, $t \in [0,1]$, and $D(\cdot \Vert Z)$ is the Gaussian relative entropy. This extends and provides a different viewpoint on the corresponding geometric inequality proved by Eskenazis and Moschidis, namely that \begin{equation} γ\left( (1-t) K_{0} + t K_{1} \right)^{\frac{1}{n}} \geq (1-t) γ(K_{0})^{\frac{1}{n}} + t γ(K_{1})^{\frac{1}{n}}, \end{equation} when $K_{0}, K_{1} \subseteq \mathbb{R}^n$ are origin-symmetric convex bodies. As an application, using Donsker--Varadhan duality, we obtain Gaussian Borell--Brascamp--Lieb inequalities applicable to even log-concave functions, which serve as functional forms of the Eskenazis--Moschidis inequality. |
| title | Entropic and functional forms of the dimensional Brunn--Minkowski inequality in Gauss space |
| topic | Probability Functional Analysis Metric Geometry 37C10, 94A17, 52A40, 52A20, 49Q22 |
| url | https://arxiv.org/abs/2504.03114 |