Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
Riemannian metrics on convex sets with applications to Poincaré and log-Sobolev inequalities
复制标题
凸集上的黎曼度量及其在庞加莱和对数索博列夫不等式中的应用
DOI：
10.1007/s00526-016-1018-3
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发表时间：
2015
影响因子：
2.1
通讯作者：
E. Milman
中科院分区：
文献类型：
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作者：
A. Kolesnikov;E. Milman
给定欧氏空间$$（\mathbb {R}^d，g_0）$$（Rd，g 0）的凸子集$$\Omega $$Ω上支持的概率测度$$\mu $$μ，我们有兴趣得到$$（\Omega，g_0，\mu）$$（Ω，g 0，μ）上的Poincaré和log-Sobolev型不等式.为此，我们将度量$$g_0$$g0改为更一般的黎曼度量g，在某种意义上适应于$$\mu $$μ，并对$$（\Omega，g，\mu）$$（Ω，g，μ）进行分析。我们考虑的度量类型是海森度量（与相关的最优运输问题密切相关），乘积度量（当$$\mu $$μ是无条件的时非常有用，即在关于坐标超平面的反射下不变），以及与欧几里得度量共形的度量，这些度量以前没有在本文中进行过探索。关于$$（\Omega，g，\mu）$$（Ω，g，μ）工具，如Brascamp-Lieb不等式的Riemann推广和Bakry-Émery准则，并通过回到原始的欧氏度量，我们得到了$$上的各种加权不等式（\Omega，g_0，\mu）$$（Ω，g0，μ）：Brascamp-Lieb不等式、加权Poincaré不等式和log-Sobolev不等式、Hardy型不等式、我们分析的关键是相关的Lichnerowicz-Bakry-Émery广义Ricci曲率张量的正性，以及流形$$（\Omega，g，\mu）$$（Ω，g，μ）的凸性。在某些情况下，我们只能确保后一个流形是（广义）平均凸的，从而在我们的不等式中产生额外的边界项。
Given a probability measure $$\mu $$μ supported on a convex subset $$\Omega $$Ω of Euclidean space $$(\mathbb {R}^d,g_0)$$(Rd,g0), we are interested in obtaining Poincaré and log-Sobolev type inequalities on $$(\Omega ,g_0,\mu )$$(Ω,g0,μ). To this end, we change the metric $$g_0$$g0 to a more general Riemannian one g, adapted in a certain sense to $$\mu $$μ, and perform our analysis on $$(\Omega ,g,\mu )$$(Ω,g,μ). The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when $$\mu $$μ is unconditional, i.e. invariant under reflections with respect to the coordinate hyperplanes), and metrics conformal to the Euclidean one, which have not been previously explored in this context. Invoking on $$(\Omega ,g,\mu )$$(Ω,g,μ) tools such as Riemannian generalizations of the Brascamp–Lieb inequality and the Bakry–Émery criterion, and passing back to the original Euclidean metric, we obtain various weighted inequalities on $$(\Omega ,g_0,\mu )$$(Ω,g0,μ): refined and entropic versions of the Brascamp–Lieb inequality, weighted Poincaré and log-Sobolev inequalities, Hardy-type inequalities, etc. Key to our analysis is the positivity of the associated Lichnerowicz–Bakry–Émery generalized Ricci curvature tensor, and the convexity of the manifold $$(\Omega ,g,\mu )$$(Ω,g,μ). In some cases, we can only ensure that the latter manifold is (generalized) mean-convex, resulting in additional boundary terms in our inequalities.