On Talagrand's Convexity Conjecture

2026/05/11 by Dongming Merrick Hua, Antoine Song, Stefan Tudose · 2 voices
Mathematics · #math.PR #math.CO #math.MG

paper · pdf

arxiv published 2026/05/11 · arxiv updated 2026/07/10

Abstract

We prove that any random vector in ℝⁿ which is dominated in convex order by a standard Gaussian vector can be written as the sum of three standard Gaussian vectors. This implies that any 1-subgaussian random vector in ℝⁿ is the sum of a universal number of Gaussian vectors. It also solves M. Talagrand's convexity problem, which in turn implies a weak version of a combinatorial analogue to the problem.

Citations

• On the subgaussian comparison theorem
• Multi-Dimensional Martingales from Mutual Information
• Simple and Sharp Generalization Bounds via Lifting
• On creating convexity in high dimensions
• A sharp version of Talagrand's selector process conjecture and an application to rounding fractional covers
• Can we spot a fake?
• The Extreme Points of Fusions
• On the Martingale Schrödinger Bridge between Two Distributions
• On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes
• A Proof of the Kahn-Kalai Conjecture
• A Gaussian fixed point random walk
• Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
• Mass transportation and contractions

Discussions

• Talagrand's Convexity Conjecture proof proposal [hn, 2 points, 1 comments]
• arxiv.org/abs/2605.10908 [bsky, 1 points, 0 comments]