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On Banach Spaces with the Helly Approximation Property

Grigory Ivanov

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.

On Banach Spaces with the Helly Approximation Property

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.
Paper Structure (11 sections, 9 theorems, 102 equations)

This paper contains 11 sections, 9 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. Idea behind the proof, type, and the no-dimensional Carathéodory lemma
  3. Reduction to a problem in the dual space
  4. No-dimensional Carathéodory lemma
  5. Properties of the Rademacher type
  6. Proofs of no-dimensional Helly-type results
  7. Proof of the non-colorful theorem
  8. Counterexample when the dual is of trivial type
  9. A counterexample in $\ell_\infty$
  10. Spaces whose dual has only trivial type
  11. Characterization

Key Result

Theorem 1.1

Let $X$ be a Banach space such that $X^*$ has Rademacher type $p\in(1,2]$ with constant $T_p(X^*)$. Set Let $\mathcal{F}$ be a finite family of convex sets in $X$. Assume that for every choice of $k$ sets $K_1, \dots, K_k \in \mathcal{F}$ the intersection $\mathbf{B}^{}_X\cap \bigcap_{i=1}^k K_i$ is non-empty. Then there exists a point $x\in X$ such that In particular, a Banach space whose dual

Theorems & Definitions (21)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1: No-dimensional Helly
  • Theorem 1.2: Colorful no-dimensional Helly
  • Remark 1.3
  • Theorem 1.3
  • Lemma 2.1: Maurey's lemma
  • Lemma 2.2: Colorful Maurey's lemma
  • proof
  • Remark 2.1
  • ...and 11 more