Talagrands Convexity Conjecture

[franzhusch]

What is the conjecture

A compact set $A \subseteq \mathbb{R}^N$ is balanced if whenever $x \in A$ and $\lambda \in \mathbb{R}$ with $|\lambda| \leq 1$, we have $\lambda x \in A$. Let $\gamma_N$ denote the standard Gaussian measure on $\mathbb{R}^N$.

Talagrand's Convexity Problem asks: Does there exist an integer $q$ (independent of the dimension $N$) such that for all $N$ and every compact balanced set $A \subseteq \mathbb{R}^N$ with $\gamma_N(A) \geq 1/2$, one can find a convex compact set $C \subseteq A + \cdots + A$ ($q$ terms in the Minkowski sum) such that $\gamma_N(C) \geq 1/2$?"

In other words, can one always create a convex set of high Gaussian measure by taking a finite (dimension-independent) number of Minkowski sums of a balanced set with high Gaussian measure?

(This description may contain subtle errors especially on more complex problems; for exact details, refer to the sources.)

A proof is worth 1000$ (I think, see sources for details).

Sources:

• https://michel.talagrand.net/prizes/convexity.tex | Michel Talagrand, "Are all sets of positive measure essentially convex?", Operator Theory: Advances and Applications, Vol. 77, Birkhäuser, 1995, pp. 295–310

Prerequisites needed

Formalizability Rating: 1/5 (0 is best) (as of 2026-01-22)

Building blocks (from Mathlib):

• Metric.Compact for compact sets in ℝ^N
• MeasureTheory.gaussianMeasure for the Gaussian measure γ_N
• Convex for convexity
• Minkowski sum operations (basic algebraic structures)

Missing pieces:

• Formal definition of balanced set (simple: ∀ x ∈ A, ∀ λ ∈ ℝ, |λ| ≤ 1 → λ • x ∈ A)
• Wrapper for the universal quantifier over dimensions and q-fold Minkowski sums

Rating justification: All essential building blocks (compact sets, Gaussian measure, convex sets, Minkowski sums) exist in Mathlib. The definition of balanced sets is straightforward. The statement itself is expressible using existing infrastructure with only minor helper definitions needed.

AMS categories

• ams-52
• ams-28
• ams-46

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• I plan on adding this conjecture to the repository
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