The Gaussian correlation inequality for centered convex sets and the case of equality
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909898892115968 |
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| author | Nakamura, Shohei Tsuji, Hiroshi |
| author_facet | Nakamura, Shohei Tsuji, Hiroshi |
| contents | Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let $X$ be a standard Gaussian random vector in $\mathbb{R}^n$. For which convex sets $K_1,K_2 \subset \mathbb{R}^n$, are the two events $\{X\in K_1\}$ and $\{X\in K_2\}$ independent? By imposing an additional normalization that $K_1$ and $K_2$ have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when $\|X\|_{K_1}$ and $\|X\|_{K_2}$ are independent as random variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04337 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Gaussian correlation inequality for centered convex sets and the case of equality Nakamura, Shohei Tsuji, Hiroshi Functional Analysis Metric Geometry Probability 39B62, 52A40 (primary), 60E15, 60G15 (secondary) Inspired by Milman's recent observation, we prove that the Gaussian correlation inequality holds for convex sets having the same barycenter, and especially for centered ones. This gives an affirmative answer to the problem proposed by Szarek and Werner. We also characterize the equality case. The study of the equality case in the non-symmetric Gaussian correlation inequality relates to the following question: Let $X$ be a standard Gaussian random vector in $\mathbb{R}^n$. For which convex sets $K_1,K_2 \subset \mathbb{R}^n$, are the two events $\{X\in K_1\}$ and $\{X\in K_2\}$ independent? By imposing an additional normalization that $K_1$ and $K_2$ have the same barycenter, we give the necessary and sufficient conditions for this independence. The conditions also identify when $\|X\|_{K_1}$ and $\|X\|_{K_2}$ are independent as random variables. |
| title | The Gaussian correlation inequality for centered convex sets and the case of equality |
| topic | Functional Analysis Metric Geometry Probability 39B62, 52A40 (primary), 60E15, 60G15 (secondary) |
| url | https://arxiv.org/abs/2504.04337 |