Talagrand's Convexity Problem
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Statement
Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian vectors: is every centered -subgaussian random vector in the sum of a universal number of standard Gaussian vectors? Both are answered affirmatively, via the sharper statement that any random vector dominated in convex order by a standard Gaussian is the sum of three standard Gaussian vectors.
Context
Talagrand posed it in 1995 and kept restating it in his problem collections through 2026; it is also Problem 54 in Ben Green's list of 100 open problems, and Talagrand identified several connections from it into probability and combinatorics.
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The disclosure is unusually precise about which proof the model owns, and the answer is: not the published one. The first two authors, working independently of the third, reached a resolution of the subgaussian formulation on the strength of a proposition whose proof GPT-5.5 Pro generated in a conversation they link a public transcript to. The third author independently arrived at a complete proof in parallel. Comparing the two, the authors judged his route more general and conceptual, so the main body follows it and the model's proposition is preserved as Appendix B. Everything outside that appendix is stated to be human authorship. So the model produced a genuine and sufficient route to the answer, which the paper then chose not to build on.
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