论文标题
几何方法用于衡量措施
Geometric Approach For Majorizing Measures
论文作者
论文摘要
高斯工艺可以视为标准希尔伯特空间的子集，但是仍然缺乏将集合大小与凸壳的大小相关的几何理解。在这项工作中，我们通过识别给定空间$ t $及其凸壳之间的覆盖数量关系（以$ t_h $为代表）来采用几何方法。如果Space $ t $是$ \ Mathbb {r}^n $中的封闭界面的Polyhedra，我们可以评估QuickHull算法获得的空间$ t $及其凸面之间的体积比。如果空间$ t $是$ \ mathbb {r}^n $中的一个通用紧凑型物体，则首先为非convex空间建立一个更一般的反向brunn-minkowski不平等，这将有助于我们在$ t $ $ t $ t_h $ to $ t_h $ to $ to $ umie $ to $ umie $ to $ to $ umie $ to $ to $ to $ to $ to $ to $ to $ to $ to $ to $ umik to $ to $ to $ to $ to $ to $ to $ to $ to $ to $ umin的元素上的元素的平均值。如果获得了空间$ t $与空间$ t_h $之间的体积比，则还可以获得空间$ t $与空间$ t_h $之间的覆盖数比，这将用于建立主要的措施不平等。对于无限的维空间，我们表明，主要尺寸不平等的恒定$ l $可能总是存在，并且存在条件将取决于$ t $的几何特性。
Gaussian processes can be considered as subsets of a standard Hilbert space, but the geometric understanding that would relate the size of a set with the size of its convex hull is still lacking. In this work, we adopt a geometric approach to the majorizing measure problem by identifying the covering number relationships between a given space $T$ and its convex hull, represented by $T_h$. If the space $T$ is a closed bounded polyhedra in $\mathbb{R}^n$, we can evaluate the volume ratio between the space $T$ and its convex hull obtained by the Quickhull algorithm. If the space $T$ is a general compact object in $\mathbb{R}^n$ with non-empty interior, we first establish a more general reverse Brunn-Minkowski inequality for nonconvex spaces which will assist us to bound the volume of $T_h$ in terms of the volume of $T$ if $T_h$ can be acquired by the finite average of the space $T$ with respect to the Minkowski sum. If the volume ratio between the space $T$ and the space $T_h$ is obtained, the covering number ratio between the space $T$ and the space $T_h$ can also be obtained which will be used to build majorizing measure inequality. For infinite dimensional space, we show that the constant $L$ at majorizing measure inequality may not always exist and the existence condition will depend on geometric properties of $T$.
