On Talagrand's Convexity Conjecture

TL;DR

This paper proves a decomposition of centered 1-subgaussian vectors into Gaussian vectors, solving Talagrand's convexity conjecture and its combinatorial analogue.

Contribution

It provides a universal decomposition result for subgaussian vectors, confirming Talagrand's convexity conjecture.

Findings

Centered 1-subgaussian vectors can be expressed as sums of a universal number of Gaussian vectors.

The result confirms Talagrand's convexity conjecture.

Implication for a related combinatorial problem.

Abstract

We prove that any centered $1$-subgaussian random vector in $\mathbb{R}^{n}$ can be written as the sum of a universal number of standard Gaussian vectors. Following the work of the second-named author, this solves M. Talagrand's convexity problem, which in turn implies a combinatorial analogue of the problem.