OP-01038 Solution claimed

Talagrand's convexity problem for Gaussian measure

Let $\gamma_\infty$ be the normalised Gaussian measure on $\mathbb{R}^{\mathbb{N}}$, and call $K \subset \mathbb{R}^{\mathbb{N}}$ balanced if $\lambda K \subset K$ whenever $|\lambda| \le 1$. Talagrand showed that the answer is no with $2K$ in place of $10K$; the constants $0.99$, $0.01$ and $10$ stand for fixed constants.

Problem. Let $K \subset \mathbb{R}^{\mathbb{N}}$ be a balanced compact set with $\gamma_\infty(K) \ge 0.99$. Must $10K$ contain a compact convex set $C$ with $\gamma_\infty(C) \ge 0.01$?

Source: M. Talagrand, Are all sets of positive measure essentially convex? (1995), and again in Are many small sets explicitly small? (STOC 2010), with a prize of 1000 dollars. Listed as Problem 54 in B. Green, 100 open problems (version of January 2026).

Imported from people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Classification

Primary: PR — Probability
Secondary: FA, MG
Keywords: convex sets, Gaussian measure, infinite-dimensional probability, Minkowski sums

References

@incollection{talagrand1995convex,
  author = {Talagrand, Michel},
  title = {Are all sets of positive measure essentially convex?},
  booktitle = {Geometric Aspects of Functional Analysis (Israel Seminar 1992--94)},
  series = {Operator Theory: Advances and Applications},
  volume = {77},
  pages = {295--310},
  publisher = {Birkh{\"a}user},
  year = {1995},
  doi = {10.1007/978-3-0348-9090-8_25}
}

@inproceedings{talagrand2010small,
  author = {Talagrand, Michel},
  title = {Are many small sets explicitly small?},
  booktitle = {Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC '10)},
  pages = {13--36},
  year = {2010},
  doi = {10.1145/1806689.1806693}
}

@misc{song2026gaussian,
  author = {Song, Antoine},
  title = {Sum of {G}aussian vectors and large sets},
  year = {2026},
  eprint = {2602.22342},
  archivePrefix = {arXiv}
}

@misc{hua2026talagrand,
  author = {Hua, Dongming Merrick and Song, Antoine and Tudose, Stefan},
  title = {On {T}alagrand's convexity conjecture},
  year = {2026},
  eprint = {2605.10908},
  archivePrefix = {arXiv}
}

@misc{mazhar2026constructive,
  author = {Mazhar, Othmane},
  title = {A constructive solution to {T}alagrand's {G}aussian convexification problem},
  year = {2026},
  eprint = {2609.14832},
  archivePrefix = {arXiv}
}

@misc{green100open,
  author = {Green, Ben},
  title = {100 open problems},
  howpublished = {\url{https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf}},
  note = {Problem 54; version of January 2026, most recent update December 2025},
  year = {2026}
}

Resolution claims

On Talagrand's Convexity Conjecture v1 arXiv ↗ AI-Verified

Dongming Merrick Hua; Antoine Song; Stefan Tudose

We prove that any random vector in $\mathbb{R}^n$ 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 $\mathbb{R}^n$ is the sum of a universal number of Gaussian vectors. It also solves M. Talagrand's convexity problem, which in turn implies …

Full resolution · not verified · fidelity not established · posted on arXiv 10 Jul 2026

Status history

When	Change	By	Reason
2026-10-29	Needs verification → Solution claimed	—	Recorded here: arXiv:2605.10908v2.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-01038, "Talagrand's convexity problem for Gaussian measure", clopen.solutions.