各向同性对数凹测度的庞加莱常数的无维数界
A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures
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中文总结 AI 辅助
本文通过三步证明方法（积分算子曲率估计、算子引理和随机定位）建立了无维数庞加莱不等式，从而证明了 KLS 猜想，并给出了欧几里得 Cheeger 常数的普遍正下界。
中文摘要 AI 辅助
Kannan--Lovász--Simonovits (KLS) 猜想询问各向同性对数凹概率测度是否满足一个与维数无关的庞加莱不等式。在本文中，我们建立了一个无维数的庞加莱不等式，从而为欧几里得 Cheeger 常数提供了一个普遍的正下界，由此证明了 KLS 猜想。我们的证明分为三步。首先，我们研究逆转微分的积分算子，并证明一个在张量指标数量上一致的曲率估计。其次，一个算子引理将低次多项式的界转化为任意次积分的界，并带有一个共同的乘法因子。第三，我们使用随机定位将这些积分界转移回多项式范数。
英文摘要
The Kannan--Lovász--Simonovits (KLS) conjecture asserts that isotropic log-concave probability measures have Poincaré constants bounded by a universal constant, independently of dimension. We give a deterministic variational proof with an explicit bound on the Poincaré constant $C_P(μ)\le25$, where $μ$ is any isotropic log-concave probability measure. Starting from elliptic moment estimates and a quadratic variance inequality, we establish geometric bounds on normalized Appell coefficient norms through \emph{joint} maximization over the measure and test function. Variation of the measure gives a maximum-principle inequality, while stationarity in the test function controls the highest-order cumulant terms. Two concave barriers constructed from quadratic and cubic polynomials close the induction. A weighted Helmholtz--Hodge decomposition then controls the curl correction of weighted divergence, yielding a curvature estimate for compatible symmetric tensor fields that is uniform in rank. Quadratic duality converts this estimate into an operator comparison for centered integration, linking the coefficient bounds to control of the inverse gradient. A spectral-radius estimate and a scalar growth inequality for adjoint iterates then yield the Poincaré bound. The final estimate is independent of the auxiliary positive curvature, allowing approximation to complete the proof for general isotropic log-concave measures.
发表机构
- Amazon（亚马逊）
- Department of Statistics, University of California, Davis（加州大学戴维斯分校统计系）
机构由 AI 辅助整理，请以论文原文为准。
