On Talagrand's Convexity Conjecture

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.

• The paper proves every centered 1-subgaussian random vector decomposes as a sum of three independent standard Gaussians, affirming Talagrand’s conjecture.
• It employs convex order domination, martingale coupling, and subgaussian comparison techniques to establish the universal decomposition.
• The results link high-dimensional probability and combinatorial covering, offering new insights into convexity extraction in Gaussian spaces.

Resolution of Talagrand's Convexity Conjecture via Gaussian Decomposition

Overview of Talagrand's Convexity Conjecture

The paper addresses a longstanding open problem posed by Talagrand concerning convexity properties of sets in high-dimensional Gaussian spaces. Specifically, the conjecture asks whether, for any closed set $A$ in $\mathbb{R}^{n}$ with large Gaussian measure ($\gamma_n(A) \geq 2/3$), there exists a convex body $K$ contained within a fixed number $q$ of Minkowski sums of $A$ such that $\gamma_n(K) \geq 1/2$. The geometric intuition is whether convexity can be "extracted" from a large set in a dimension-agnostic way via repeated Minkowski summation, despite measure concentration phenomena and the lack of geometric convexity of large sets.

Notably, Talagrand demonstrated that $q=2$ is insufficient—even allowing rescaling—and that the problem is false under certain convexity operations or strengthening. Until this work, partial progress linked the conjecture to properties of subgaussian random vectors and combinatorial analogues related to subset covering problems. The equivalence between the geometric Minkowski-sum convexity problem and a structural characterization of subgaussian random vectors was established in prior work.

Main Results and Numerical Outcomes

The central contribution is the proof that any centered $1$-subgaussian random vector in $\mathbb{R}^{n}$ can be written as the sum of a universal constant ($\mathbb{R}^{n}$0) number of standard Gaussian vectors, i.e.: $\mathbb{R}^{n}$1 where $\mathbb{R}^{n}$2 are independent standard Gaussian random vectors. This result affirms Talagrand's conjecture, resolving both the geometric and probabilistic formulations.

Contrasting previous lower bounds, this result demonstrates that the universal decomposition holds for $\mathbb{R}^{n}$3 and that two Gaussians are insufficient (as shown by canonical examples), marking a strong numerical threshold. The proof hinges on convex order domination, subgaussian comparison theorems, and martingale coupling constructions leveraging log-concave distributions and entropy maximization principles.

An immediate combinatorial consequence is the resolution of Talagrand's combinatorial analogue: For any subset $\mathbb{R}^{n}$4 of $\mathbb{R}^{n}$5 with product measure at least $\mathbb{R}^{n}$6, the set of points not covered by $\mathbb{R}^{n}$7 elements of $\mathbb{R}^{n}$8 is explicitly small—quantified in terms of $\mathbb{R}^{n}$9 for $\gamma_n(A) \geq 2/3$0-small sets—in the sense of measure-theoretic covering.

Technical Approach

The authors use several key analytic and probabilistic tools:

• Convex Order Domination: The equivalence (via Strassen's Theorem [strassen]) between domination in convex order and the existence of martingale couplings enables the reduction from the geometric Minkowski-sum problem to the probabilistic sum-of-Gaussians formulation.
• Subgaussian Comparison Theorem: Van Handel's result [Van25] shows that for any centered $\gamma_n(A) \geq 2/3$1-subgaussian $\gamma_n(A) \geq 2/3$2, $\gamma_n(A) \geq 2/3$3 is convex-order dominated by a standard Gaussian $\gamma_n(A) \geq 2/3$4 for universal $\gamma_n(A) \geq 2/3$5.
• Martingale Coupling via Entropy Maximization: The coupling construction involves maximizing conditional entropy under the convex order constraint, revealing that extremal solutions correspond to log-concave densities of specific exponential-affine form.
• Reduction to Finitely Supported Random Vectors: By Prokhorov’s theorem, the class of random vectors decomposable as sums of Gaussians is closed under weak limits, facilitating the reduction in proofs.
• Combinatorial Translation: The solution to the geometric problem implies the combinatorial covering result, bridging additive and probabilistic combinatorics.

Numerically, the universal constant $\gamma_n(A) \geq 2/3$6 is sharp, and the bounds $\gamma_n(A) \geq 2/3$7 for $\gamma_n(A) \geq 2/3$8, with convex body measure at least $\gamma_n(A) \geq 2/3$9, are attained via the decomposition and coupling arguments.

Implications and Connections

The resolution establishes a novel structural decomposition of subgaussian random vectors and characterizes high-dimensional convexity phenomena under Gaussian measure. The result strengthens Talagrand’s earlier comparison theorems, providing a precise sum-of-Gaussians representation.

Practically, the findings impact stochastic geometry, additive combinatorics, and high-dimensional probability, particularly in understanding how convexity can be synthesized from non-convex large sets. The combinatorial analogue is closely linked to the Kahn-Kalai conjecture, recently resolved, and selector process conjectures relevant in empirical process theory.

Theoretically, the solution reveals unexpected connections between martingale optimal transport, convex order, and entropy methods, suggesting new directions for convexification techniques, measure concentration, and stochastic process analysis.

Future Directions

Potential avenues include:

• Refinement of constants, explicit constructions, and tightening “convexification” rates under various measures.
• Exploration of generalizations to non-Gaussian settings, log-concave measures, and more complex metric spaces.
• Applications to empirical risk minimization, sparsity, and high-dimensional statistical inference where convex hulls play a role.
• Interdisciplinary connections to optimal transport, entropy-regularized stochastic flows, and combinatorics.

Progress in semi-discrete optimal transport and Laguerre tessellation fitting, as alluded to in the appendix, may further elucidate the structural aspects of convex domination and martingale couplings.

Conclusion

By proving that any centered subgaussian random vector can be decomposed as the sum of three independent standard Gaussian vectors, the paper affirmatively resolves Talagrand's convexity problem and its combinatorial analogues. The result sharpens our understanding of convexity in Gaussian spaces and connects probabilistic inequalities, geometric measure theory, and combinatorial covering principles. These advances open further opportunities in high-dimensional analysis, probability, and additive combinatorics.