The Kottman constant for $α$-Hölder maps

TL;DR

This work analyzes how the geometric Kottman constant $\kappa(X)$ governs the extendability of $\alpha$-Hölder maps from subsets of a Banach space $X$ into sequence spaces. It extends Kalton's Lipschitz-extension framework to $\alpha$-Hölder maps by proving the exact equality $\lambda_\alpha(X,c_0)=\kappa(X)^\alpha$ for all infinite-dimensional $X$ and all $\alpha\in(0,1]$, leveraging the snowflake metric $X^\alpha$. It also provides a quantitative lower bound for extensions into $L_q$ spaces when $1<q\le2$, namely $2^{-1/q^*}\kappa(X)^\alpha\le\lambda_\alpha(X,L_q)$, derived via Naor's estimates with Rademacher functions, with concrete implications for extensions between $L_p$ and $L_q$. These results clarify geometric obstructions to Hölder extensions and give precise constants that tie extension behavior to the Kottman constant, enhancing understanding of when isometric or Lipschitz-type extensions are possible. The findings have implications for the structure of infinite-dimensional Banach spaces and the feasibility of controlled Hölder extensions to common sequence spaces.

Abstract

We investigate the role of the Kottman constant of a Banach space $X$ in the extension of $α$-Hölder continuous maps for every $α\in (0,1]$.

Theorems & Definitions (13)

• Proposition 1
• Lemma 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Corollary 1
• Corollary 2
• Lemma 2