基于高斯拟合和自由能估计的凸区域Bobkov谱界的证明
A Proof of Bobkov's Spectral Bound For Convex Domains via Gaussian Fitting and Free Energy Estimation
摘要：
本文给出了欧几里得空间中有界凸区域$K$上Neumann Laplacian（负）第一正特征值（或Cheeger常数）的Bobkov下界的一个新证明.我们的证明避免使用本地化方法或任何几何扩展。相反，我们推导出的下界调用一个谱转移原则的日志凹措施，比较$K$上的均匀测量与适当的缩放高斯测量的条件是$K$。争论的关键是要在这两个测度（比如相对熵或总变差距离）之间建立一个良好的重叠，这归结为获得条件高斯测度自由能的精确下界。
英文摘要：
We obtain a new proof of Bobkov's lower bound on the first positive eigenvalue of the (negative) Neumann Laplacian (or equivalently, the Cheeger constant) on a bounded convex domain $K$ in Euclidean space. Our proof avoids employing the localization method or any of its geometric extensions. Instead, we deduce the lower bound by invoking a spectral transference principle for log-concave measures, comparing the uniform measure on $K$ with an appropriately scaled Gaussian measure which is conditioned on $K$. The crux of the argument is to establish a good overlap between these two measures (in say the relative-entropy or total-variation distances), which boils down to obtaining sharp lower bounds on the free energy of the conditioned Gaussian measure.
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