No-dimensional Helly and dual Maurey’s lemma in cotype q Banach spaces

Determine whether Banach spaces of cotype q ≥ 2 admit analogues of both (i) a no-dimensional Helly-type theorem—namely, the existence of a sequence r_k(X) > 0 with r_k(X) → 0 such that, whenever K_1, …, K_n are convex subsets of X whose every k-wise intersection meets the unit ball, there exists a point x ∈ X with dist(x, K_i) ≤ r_k(X) for all i—and (ii) a corresponding dual version of Maurey’s lemma formulated for Banach spaces of cotype q ≥ 2.

Background

This work establishes no-dimensional Helly-type theorems (including fractional and colorful variants) in uniformly convex Banach spaces, with bounds expressed via the modulus of convexity. The approach adapts the Euclidean arguments of Adiprasito–Bárány–Mustafa–Terpai by replacing the Pythagorean theorem with inequalities depending on the modulus of convexity.

The paper emphasizes a duality between no-dimensional Helly and no-dimensional Carathéodory theorems. Prior results indicate that no-dimensional Carathéodory-type statements naturally align with smoothness (type) conditions, while the present Helly results align with convexity (uniform convexity) conditions. Motivated by probabilistic methods that yield additional corollaries in spaces of type p > 1, the author asks whether analogous no-dimensional Helly and dual Maurey-type results can be developed in Banach spaces characterized by cotype q ≥ 2.