Improved bounds in the scaled Enflo type inequality for Banach spaces

TL;DR

This paper improves bounds related to the scaled Enflo type inequality in Banach spaces with Rademacher type p, refining previous results by employing an enhanced smoothing and approximation technique.

Contribution

It introduces a tighter bound on m in the scaled Enflo type inequality for Banach spaces, advancing the understanding of metric embeddings.

Findings

Bound m < Cn^{2-1/p}log n established

Improved over previous bound m < Cn^{3-2/p}

Enhanced smoothing and approximation scheme used

Abstract

It is shown that if (X,||.||_X) is a Banach space with Rademacher type p \ge 1, then for every integer n there exists an even integer m < Cn^{2-1/p}log n (C is an absolute constant), such that for every f:Z_m^n --> X, \Avg_{x,\e}[||f(x+ m\e/2)-f(x)}||_X^p] < C(p,X) m^p\sum_{j=1}^n\Avg_x[||f(x+e_j)-f(x)||_X^p], where the expectation is with respect to uniformly chosen x \in Z_m^n and \e \in \{-1,1\}^n, and C(p,X) is a constant that depends on p and the Rademacher type constant of X. This improves a bound of m < Cn^{3-2/p} that was obtained in [Mendel, Naor 2007]. The proof is based on an augmentation of the "smoothing and approximation" scheme, which was implicit in [Mendel, Naor 2007].