论文标题
超出尼尔森的时间及其在blaschkekke-santaló的不平等和倒数santaló不平等现象之外的超出费用。
Hypercontractivity beyond Nelson's time and its applications to Blaschke--Santaló inequality and inverse Santaló inequality
论文作者
论文摘要
我们探讨了诸如Ornstein流量（uhlenbeck流量和fokker）的扩散流的分析之间的相互作用 - 泛力流量与凸几何相关的不等式。更确切地说，我们引入了Ornstein的新型超收缩率 - uhlenbeck流动，并阐明这些意味着Blaschkeke-Santaló-santaló不平等和逆Santaló不平等现象，也称为Mahler的猜想。在本文中，我们建立了两种新的超收缩率。第一个是在尼尔森的时间关系上，在输入具有适当的对称性的限制下，Borell反向超收缩不平等的改善。然后，我们证明这意味着Blaschke-Santaló不平等。同时，它还提供了一个逆Brascamp- lieb不平等的示例 - 沃尔夫 - 沃尔夫（Wolff）超出了其非分类状态。第二个是尼尔森的正向超收缩不平等，指数低于1的指数，即log-convex和semi-log-concave。这产生了凸界弯曲良好的凸体的凸体产物的新下限。这种后果为Stancu和Reisner--Schütt的作品提供了定量结果 - 他们观察到具有弯曲良好边界的凸体不是体积产物的局部最小值。
We explore an interplay between an analysis of diffusion flows such as Ornstein--Uhlenbeck flow and Fokker--Planck flow and inequalities from convex geometry regarding the volume product. More precisely, we introduce new types of hypercontractivity for the Ornstein--Uhlenbeck flow and clarify how these imply the Blaschke--Santaló inequality and the inverse Santaló inequality, also known as Mahler's conjecture. Motivated the link, we establish two types of new hypercontractivity in this paper. The first one is an improvement of Borell's reverse hypercontractivity inequality in terms of Nelson's time relation under the restriction that the inputs have an appropriate symmetry. We then prove that it implies the Blaschke--Santaló inequality. At the same time, it also provides an example of the inverse Brascamp--Lieb inequality due to Barthe--Wolff beyond their non-degenerate condition. The second one is Nelson's forward hypercontractivity inequality with exponents below 1 for the inputs which are log-convex and semi-log-concave. This yields new lower bounds of the volume product for convex bodies whose boundaries are well curved. This consequence provides a quantitative result of works by Stancu and Reisner--Schütt--Werner where they observed that a convex body with well curved boundary is not a local minimum of the volume product.
