On creating convexity in high dimensions

Given a subset $A$ of ${\mathbb{R}}^n$, we define \begin{align*} $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}$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 ${\gamma }_n$ denote the standard Gaussian measure on ${\mathbb{R}}^n$. We show that for every $\epsilon >0$, there exists a subset $A$ of ${\mathbb{R}}^n$ with Gaussian measure ${\gamma }_n(A)\ge 1-\epsilon$ such that for all $k=O_{\epsilon }(\sqrt{\log \log (n)})$, ${\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}}_k(A)$ contains no convex set $K$ of Gaussian measure ${\gamma }_n(K)\ge \epsilon$. 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.