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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.07981 |
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| _version_ | 1866912888426332160 |
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| author | Milman, Emanuel Nakamura, Shohei Tsuji, Hiroshi |
| author_facet | Milman, Emanuel Nakamura, Shohei Tsuji, Hiroshi |
| contents | We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $γ$ on $\mathbb{R}^n$, stating that \[ γ(K) γ(L) \leq γ(K\cap L) γ(K+L) \] whenever $K$ and $L$ are origin-symmetric convex sets in $\mathbb{R}^n$. This confirms a conjecture of M. Tehranchi [https://doi.org/10.1214/17-ECP89]. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of $K$ and $L$ are at the origin, and characterize the equality cases. After rescaling, this also yields the following new inequality for convex sets with (Lebesgue) barycenters at the origin: \[ |K| |L| \leq |K \cap L| |K + L | ; \] this can be seen as a conjugate counterpart to Spingarn's extension of the Rogers-Shephard inequality (where $K+L$ is replaced by $K-L$ above). We also derive an additional conjugate version of a Gaussian inequality due to V. Milman and Pajor, as well as several extensions. Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_07981 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Gaussian Conjugate Rogers-Shephard Inequality Milman, Emanuel Nakamura, Shohei Tsuji, Hiroshi Functional Analysis Probability We fuse between the Rogers-Shephard inequality for the Lebesgue measure and Royen's Gaussian Correlation Inequality, simultaneously extending both into a single sharp inequality for the Gaussian measure $γ$ on $\mathbb{R}^n$, stating that \[ γ(K) γ(L) \leq γ(K\cap L) γ(K+L) \] whenever $K$ and $L$ are origin-symmetric convex sets in $\mathbb{R}^n$. This confirms a conjecture of M. Tehranchi [https://doi.org/10.1214/17-ECP89]. In fact, we show that the inequality remains valid whenever the Gaussian barycenters of $K$ and $L$ are at the origin, and characterize the equality cases. After rescaling, this also yields the following new inequality for convex sets with (Lebesgue) barycenters at the origin: \[ |K| |L| \leq |K \cap L| |K + L | ; \] this can be seen as a conjugate counterpart to Spingarn's extension of the Rogers-Shephard inequality (where $K+L$ is replaced by $K-L$ above). We also derive an additional conjugate version of a Gaussian inequality due to V. Milman and Pajor, as well as several extensions. Our main tool is a new Gaussian Forward-Reverse Brascamp-Lieb inequality for centered log-concave functions, of independent interest, which is crucially applicable to degenerate Gaussian covariances. |
| title | The Gaussian Conjugate Rogers-Shephard Inequality |
| topic | Functional Analysis Probability |
| url | https://arxiv.org/abs/2602.07981 |