Gaussian Width of Convex Sets via Integral Decompositions, Projections, and the Distribution of Intrinsic Volumes

TL;DR

This paper introduces new geometric decompositions for bounding the Gaussian width of convex sets, connecting intrinsic volumes, local widths, and metric projections without relying on generic chaining, advancing understanding of Gaussian processes.

Contribution

It develops two novel decompositions of Gaussian width based on the geometry of convex sets, linking intrinsic volumes and local widths without generic chaining.

Findings

Bounds Gaussian width using intrinsic volumes and local metric structure.

Connects Wills functional with Gaussian widths via intrinsic volumes.

Recovers a local Dudley integral in the worst case.

Abstract

We revisit the problem of bounding the expected supremum of a canonical Gaussian process indexed by a convex set $T \subset \mathbf{R}^d$. We develop two decompositions for the Gaussian width, based on the geometry of the index set. The first decomposition involves metric projections of Gaussians onto rescaled copies of $T$. The second involves fixed points arising from a quadratically penalized variant of the local width. Neither decomposition directly invokes generic chaining constructions. Our results make use of recent work in geometric analysis and Gaussian processes. The work of Chatterjee [Ann. Statist., 2014] characterizes the behavior of the metric projection of a Gaussian random vector onto rescaled copies of $T$ with a variational problem involving localized Gaussian widths. We use these bounds to develop decompositions of the Gaussian width using the local metric structure of $T$. Second, we leverage the work of Vitale [Ann. Probab., 1996] to form a connection between the Wills functional (and hence the intrinsic volumes of $T$) and the first terms that appear in our decompositions. Finally, invoking recent work by Mourtada [J. Eur. Math. Soc., 2025] on the logarithm of the Wills functional, we show that the width is controlled by a single, ''peak index'' of the intrinsic volumes. In the worst case, our bound recovers a local form of the classical Dudley integral.