# Variance estimates and almost Euclidean structure

## TL;DR

This paper introduces new parameters for norms and log-concave measures that yield sharp inequalities, offering insights into the structure of high-dimensional spaces and providing a novel proof of Dvoretzky's theorem.

## Contribution

It develops new parameters related to norms and measures that enhance understanding of high-dimensional geometry and offers a concise proof of Dvoretzky's theorem.

## Key findings

- New parameters provide sharp distributional inequalities.
- Insights into the local structure of high-dimensional spaces.
- A short proof of Dvoretzky's theorem is presented.

## Abstract

We introduce and initiate the study of new parameters associated with any norm and any log-concave measure on $\mathbb R^n$, which provide sharp distributional inequalities. In the Gaussian context this investigation sheds light to the importance of the statistical measures of dispersion of the norm in connection with the local structure of the ambient space. As a byproduct of our study, we provide a short proof of Dvoretzky's theorem which not only supports the aforementioned significance but also complements the classical probabilistic formulation.