On Banach Spaces with the Helly Approximation Property

TL;DR

This paper characterizes Banach spaces with the Helly approximation property, linking it to the dual space having non-trivial Rademacher type, and provides a colorful version with average radius control.

Contribution

It establishes an equivalence between the Helly approximation property and the dual space's Rademacher type, introducing a new duality-based proof and a colorful variant.

Findings

Helly approximation property holds iff the dual space has non-trivial Rademacher type

Introduces a duality argument using Maurey's empirical method

Provides a colorful version with control over average radii

Abstract

Qualitatively, a no-dimensional Helly-type theorem says that if every small subfamily of convex sets has a common point in a bounded region, then suitable neighborhoods of all the sets in the whole family have a common point. Quantitative bounds, when available, depend on the ambient metric. We say that a Banach space has the Helly approximation property if the radii of these neighborhoods tend to zero as the size of the subfamilies tends to infinity. In this paper, we show that the Helly approximation property holds if and only if the dual space has non-trivial Rademacher type. The argument combines Maurey's empirical method with a duality argument at a minimizer of the maximal distance function. We also prove a colorful version of this theorem, with control over the average of the radii.