Geometric Approach For Majorizing Measures

TL;DR

This paper introduces a geometric approach to the majorizing measure problem for Gaussian processes, relating set sizes to convex hulls using volume ratios and covering numbers, with implications for infinite-dimensional spaces.

Contribution

It develops a geometric framework linking covering numbers and volume ratios for convex and non-convex sets, extending the understanding of majorizing measures in Gaussian processes.

Findings

Established a reverse Brunn-Minkowski inequality for nonconvex spaces.

Derived volume ratio bounds between a set and its convex hull.

Identified conditions affecting the existence of the majorizing measure constant in infinite dimensions.

Abstract

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 $T$.