A constructive solution to Talagrand's Gaussian convexification problem
Abstract
Talagrand asked for a construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. We first prove a stronger nonsymmetric existential statement: if A⊂ℝn is measurable and γn(A)>2/3, then A+A+A contains a compact convex set of Gaussian measure at least 1/2. Let now A be closed with γn(A)≥7/8, let Φ be the standard Gaussian distribution function, and put a0=Φ−1(3/4). For every Λ>1 and every 0<p<2Φ(Λa0)−1 we construct a centrally symmetric finite polytope C⊂Λ(A+A+A) with γn(C)≥ p. Thus measure 3/4 is obtained for every Λ>1.705510…. We give matching upper and lower bounds for the optimal high-measure dilation and show that the dilation profile used by the construction is sharp among all symmetric half-measure cores. Finally, if γn(A)≥5/6+η, we construct, for every 0<ε≤1/2, a centered ellipsoid Eε with γn(Eε)≥1−ε and c/(Φ−1(1−ε/2))√((log n)/n) Eε⊂ A+A+A. Both the dimension dependence and the dependence on ε are optimal up to constants. The construction also gives a six-summand theorem for balanced sets and a second finite construction based on subgaussian tests.
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BibTeX
@misc{mazhar2026constructive,
title = {{A constructive solution to Talagrand's Gaussian convexification problem}},
author = {Othmane Mazhar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14832},
eprint = {2609.14832},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14832},
}
RIS
TY - UNPB
TI - A constructive solution to Talagrand's Gaussian convexification problem
AU - Othmane Mazhar
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.14832
UR - https://arxiv.org/abs/2609.14832
AB - Talagrand asked for a construction of a large convex subset of a bounded Minkowski sum of a large Gaussian set. We first prove a stronger nonsymmetric existential statement: if A⊂ℝ^n is measurable and γ_n(A)>2/3, then A+A+A contains a compact convex set of Gaussian measure at least 1/2. Let now A be closed with γ_n(A)≥7/8, let Φ be the standard Gaussian distribution function, and put a_0=Φ^−1(3/4). For every Λ>1 and every 0<p<2Φ(Λa_0)−1 we construct a centrally symmetric finite polytope C⊂Λ(A+A+A) with γ_n(C)≥ p. Thus measure 3/4 is obtained for every Λ>1.705510…. We give matching upper and lower bounds for the optimal high-measure dilation and show that the dilation profile used by the construction is sharp among all symmetric half-measure cores. Finally, if γ_n(A)≥5/6+η, we construct, for every 0<ε≤1/2, a centered ellipsoid E_ε with γ_n(E_ε)≥1−ε and c/(Φ^−1(1−ε/2))√((log n)/n) E_ε⊂ A+A+A. Both the dimension dependence and the dependence on ε are optimal up to constants. The construction also gives a six-summand theorem for balanced sets and a second finite construction based on subgaussian tests.
ER -