Dichotomies, structure, and concentration in normed spaces

TL;DR

This paper establishes a probabilistic deviation inequality for norms in finite-dimensional normed spaces, leading to improved bounds on the existence of nearly Euclidean subspaces with dimensions proportional to the logarithm of the space's dimension.

Contribution

It introduces a new deviation inequality using probabilistic and combinatorial methods, enhancing previous results on Euclidean subspace concentration in normed spaces.

Findings

Proves a new deviation inequality for normed spaces involving Gaussian vectors.

Shows existence of large nearly Euclidean subspaces with improved bounds.

Enhances previous results by a logarithmic factor in epsilon.

Abstract

We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space $X=(\mathbb R^n ,\|\cdot\| )$ there exists an invertible linear map $T:\mathbb R^n \to \mathbb R^n$ with \[ \mathbb P\left( \big| \|TG\| -\mathbb E\|TG\| \big| > \varepsilon \mathbb E\|TG\| \right) \leq C\exp \left( -c\max\{ \varepsilon^2, \varepsilon \} \log n \right),\quad \varepsilon>0, \] where $G$ is the standard $n$-dimensional Gaussian vector and $C,c>0$ are universal constants. It follows that for every $\varepsilon\in (0,1)$ and for every normed space $X=(\mathbb R^n,\|\cdot\|)$ there exists a $k$-dimensional subspace of $X$ which is $(1+\varepsilon)$-Euclidean and $k\geq c\varepsilon \log n/\log\frac{1}{\varepsilon}$. This improves by a logarithmic on $\varepsilon$ term the best previously known result due to G. Schechtman.