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：我们重新审视了由凸集 $T \subset \mathbf{R}^d$ 索引的典范高斯过程期望上确界的边界问题。基于索引集的几何性质，我们发展了高斯宽度的两种分解。第一种分解涉及高斯向量到 $T$ 的缩放副本上的度量投影。第二种分解依赖于由局部宽度的二次惩罚变体产生的固定点。这两种分解均未直接采用一般链式构造。我们的结果利用了近期在几何分析与高斯过程领域的研究工作。Chatterjee [Ann. Statist., 2014] 的研究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 $T$ 缩放副本上度量投影的行为。我们利用这些边界，基于 $T$ 的局部度量结构发展了高斯宽度的分解。其次，我们借助 Vitale [Ann. Probab., 1996] 的工作，建立了 Wills 泛函（进而包括 $T$ 的内蕴体积）与我们分解中出现的首项之间的联系。最终，通过引用 Murtada [J. Eur. Math. Soc., 2025] 关于 Wills 泛函对数的近期工作，我们证明了高斯宽度可由内蕴体积的单一“峰值指标”控制。在最坏情况下，我们的边界恢复了经典 Dudley 积分的局部形式。