Talagrand’s creating convexity conjecture (Minkowski addition)

Determine whether, for each ε > 0, there exists a universal integer k = k_ε > 1 such that for every integer n ≥ 1 and every balanced subset A ⊆ R^n with standard Gaussian measure γ_n(A) ≥ 1 − ε, the k-fold Minkowski sum A^k := {s_1 + … + s_k : s_i ∈ A} contains a convex subset K with γ_n(K) > ε.

Background

The conjecture originates from a question of Talagrand about whether large convex subsets can be produced from a set A in high dimensions using a number of operations independent of the dimension n. Taking the standard Gaussian measure γ_n as the notion of size and Minkowski addition as the operation, the conjecture posits a dimension-free number of additions sufficient to produce a convex subset of prescribed Gaussian mass from any balanced set A of large Gaussian measure.

The present paper disproves a stronger version of this conjecture when Minkowski addition is replaced by repeated convex combinations (conv_k(A)), even allowing k to grow like O(√log log n). However, the original conjecture involving Minkowski sums (A^k) remains unresolved; the negative result here suggests any potential truth of the original conjecture would rely crucially on the dilation inherent to Minkowski addition.