On creating convexity in high dimensions

TL;DR

The paper constructs large high-dimensional sets with specific convexity properties, showing limitations of convex combinations in capturing significant Gaussian measure, thus addressing a conjecture of Talagrand.

Contribution

It demonstrates the existence of large sets in high dimensions where convex combinations up to a certain order do not contain large Gaussian measure sets, countering a conjecture of Talagrand.

Findings

Existence of large sets with limited convexity in high dimensions

Convex combinations of order up to O(√log log n) cannot capture large Gaussian measure sets

Application of optimal transport and concentration of measure techniques

Abstract

Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ \lambda_1 s_1 + \cdots + \lambda_k s_k : \lambda_i \in [0,1], \sum_{i=1}^k \lambda_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 $\gamma_n$ 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 $\gamma_n(A) \geq 1- \varepsilon$ such that for all $k = O_\varepsilon(\sqrt{\log \log(n)})$, $\mathrm{conv}_k(A)$ contains no convex set $K$ of Gaussian measure $\gamma_n(K) \geq \varepsilon$. 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.