A generalized Legendre duality relation and Gaussian saturation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913537113194496 |
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| author | Nakamura, Shohei Tsuji, Hiroshi |
| author_facet | Nakamura, Shohei Tsuji, Hiroshi |
| contents | Motivated by the barycenter problem in optimal transportation theory, Kolesnikov--Werner recently extended the notion of the Legendre duality relation for two functions to the case for multiple functions. We further generalize the duality relation and then establish the centered Gaussian saturation property for a Blaschke--Santaló type inequality associated with it. Our approach to the understanding such a generalized Legendre duality relation is based on our earlier observation that directly links Legendre duality with the inverse Brascamp--Lieb inequality. More precisely, for a large family of degenerate Brascamp--Lieb data, we prove that the centered Gaussian saturation property for the inverse Brascamp--Lieb inequality holds true when inputs are restricted to even and log-concave functions.
As an application to convex geometry, we establish the most important case of a conjecture of Kolesnikov and Werner about the Blaschke--Santaló inequality for multiple even functions as well as multiple symmetric convex bodies. Furthermore, in the direction of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture of Kolesnikov--Werner about a Talagrand type inequality for multiple even probability measures that involves the Wasserstein barycenter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_13611 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalized Legendre duality relation and Gaussian saturation Nakamura, Shohei Tsuji, Hiroshi Functional Analysis Information Theory Classical Analysis and ODEs Metric Geometry 39B62, 52A40 (primary), 26D15, 52A38, 94A17 (secondary) Motivated by the barycenter problem in optimal transportation theory, Kolesnikov--Werner recently extended the notion of the Legendre duality relation for two functions to the case for multiple functions. We further generalize the duality relation and then establish the centered Gaussian saturation property for a Blaschke--Santaló type inequality associated with it. Our approach to the understanding such a generalized Legendre duality relation is based on our earlier observation that directly links Legendre duality with the inverse Brascamp--Lieb inequality. More precisely, for a large family of degenerate Brascamp--Lieb data, we prove that the centered Gaussian saturation property for the inverse Brascamp--Lieb inequality holds true when inputs are restricted to even and log-concave functions. As an application to convex geometry, we establish the most important case of a conjecture of Kolesnikov and Werner about the Blaschke--Santaló inequality for multiple even functions as well as multiple symmetric convex bodies. Furthermore, in the direction of information theory and optimal transportation theory, this provides an affirmative answer to another conjecture of Kolesnikov--Werner about a Talagrand type inequality for multiple even probability measures that involves the Wasserstein barycenter. |
| title | A generalized Legendre duality relation and Gaussian saturation |
| topic | Functional Analysis Information Theory Classical Analysis and ODEs Metric Geometry 39B62, 52A40 (primary), 26D15, 52A38, 94A17 (secondary) |
| url | https://arxiv.org/abs/2409.13611 |