Samuel Johnston

On creating convexity in high dimensions

Given a subset A of R^n, let conv_k(A) denote the set of all k-fold convex combinations of points in A; that is, vectors of the form λ_1 a_1 + ... + λ_k a_k with a_i in A, λ_i ≥ 0, and λ_1 + ... + λ_k = 1. Carathéodory’s theorem in convex geometry states that the convex hull of A is precisely conv_{n+1}(A): in other words, every point in the convex hull can be written as a convex combination of at most n+1 points from A.
In the 1990s, Talagrand asked whether large convex subsets can already be constructed from large sets using only a bounded number k < n of convex combinations, independent of the dimension. In a Gaussian formulation: if g_n denotes the standard Gaussian measure on R^n, does there exist a universal k such that for every set A with g_n(A) > 0.9, the set conv_k(A) contains a convex subset B with g_n(B) > 0.1?
Although this problem lies in convex geometry, I will discuss how in the preprint https://arxiv.org/abs/2502.10382, ideas from probability and optimal transport provide new perspectives and some initial inroads on Talagrand's question.