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.

翻译：本文重新审视了以凸集$T \subset \mathbf{R}^d$为索引的典型高斯过程期望上确界的估计问题。基于索引集的几何结构，我们提出了两种高斯宽度的分解方法。第一种分解涉及高斯向量到$T$缩放副本上的度量投影；第二种分解则基于局部宽度的二次惩罚变体所产生的固定点。两种分解均未直接调用通用链式构造。我们的结果利用了几何分析与高斯过程领域的最新研究成果。Chatterjee [Ann. Statist., 2014] 的工作通过涉及局部高斯宽度的变分问题，刻画了高斯随机向量到$T$缩放副本的度量投影行为。我们利用这些界，结合$T$的局部度量结构，发展了高斯宽度的分解方法。其次，借助Vitale [Ann. Probab., 1996] 的研究，我们建立了Wills泛函（进而$T$的内蕴体积）与分解式中首项之间的联系。最后，基于Mourtada [J. Eur. Math. Soc., 2025] 关于Wills泛函对数的最新工作，我们证明宽度可由内蕴体积的单一“峰值指标”控制。在最坏情形下，我们的界可恢复经典Dudley积分的局部形式。