广义勒让德对偶关系与高斯饱和性
A generalized Legendre duality relation and Gaussian saturation
AI总结：
本研究推广了勒让德对偶关系，基于勒让德对偶与逆Brascamp–Lieb不等式的关联，证明了退化Brascamp–Lieb数据下偶函数与对数凹函数输入的中心化高斯饱和性，解决了凸几何及信息论、最优输运领域的两个相关猜想。
摘要：
受最优输运理论中重心问题的启发，Kolesnikov–Werner 近期将两函数的勒让德对偶关系概念推广至多函数情形。我们进一步推广了该对偶关系，并建立了与之关联的 Blaschke–Santaló 型不等式的中心化高斯饱和性质。我们理解这类广义勒让德对偶关系的方法，基于此前将勒让德对偶与逆 Brascamp–Lieb 不等式直接关联的观察。更确切地说，对于一大类退化的 Brascamp–Lieb 数据，我们证明了当输入限制为偶函数且对数凹函数时，逆 Brascamp–Lieb 不等式的中心化高斯饱和性质成立。作为在凸几何中的应用，我们证明了 Kolesnikov 和 Werner 关于多个偶函数及多个对称凸体的 Blaschke–Santaló 不等式猜想的最重要情形。此外，在信息论与最优输运理论方向，本研究对 Kolesnikov–Werner 提出的另一项猜想给出了肯定回答，该猜想涉及含 Wasserstein 重心的多个偶概率测度的 Talagrand 型不等式。
英文摘要：
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.
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