On creating convexity in high dimensions
Samuel G. G. Johnston
TL;DR
This work addresses the problem of creating convexity in high dimensions by studying $\mathrm{conv}_k(A)$, the set of points that can be formed from $k$ convex operations on a base set $A$. The authors prove a negative result for the stronger Talagrand-style conjecture: there exist large $A_n$ with $\gamma_n(A_n) \ge 1 - O(1/n)$ such that $\mathrm{conv}_k(A_n)$ contains no convex subset of substantial Gaussian measure for $k$ as small as $O_\varepsilon(\sqrt{\log\log n})$, for universal constants. The proof combines an empirical-coordinate framework with optimal transport and copula techniques, establishing upper and lower bounds on exceedances of convex sums of Gaussians and connecting these to Wasserstein-distance stability under convex combinations. The results imply that, unless one allows the Minkowski-sum dilation, a fixed small number of convex operations cannot generate large convex structures in high dimensions, illuminating intrinsic limits in high-dimensional convexity construction. The work thus advances our understanding of when large-scale convexity can emerge from large, random-like sets under dimension-dependent constraints.
Abstract
Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ λ_1 s_1 + \cdots + λ_k s_k : λ_i \in [0,1], \sum_{i=1}^k λ_i = 1 , s_i \in A \right\} \end{align*} to be the set of vectors in $\mathbb{R}^n$ that can be written as a $k$-fold convex combination of vectors in $A$. Let $γ_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. We show that for every $\varepsilon > 0$, there exists a subset $A$ of $\mathbb{R}^n$ with Gaussian measure $γ_n(A) \geq 1- \varepsilon$ such that for all $k = O_\varepsilon(\sqrt{\log \log(n)})$, $\mathrm{conv}_k(A)$ contains no convex set $K$ of Gaussian measure $γ_n(K) \geq \varepsilon$. This provides a negative resolution to a stronger version of a conjecture of Talagrand. Our approach utilises concentration properties of random copulas and the application of optimal transport techniques to the empirical coordinate measures of vectors in high dimensions.