Upper bound for the Dvoretzky dimension in Milman-Schechtman theorem

TL;DR

This paper provides an elementary proof of the upper bound for the Dvoretzky dimension in Milman-Schechtman theorem, removing previous restrictions on the body's parameters.

Contribution

It offers a simplified proof of the upper bound for Dvoretzky dimension without restrictions on the body's parameters.

Findings

Elementary proof of the upper bound for Dvoretzky dimension.

Removes restrictions on the ratio of M(K) to b(K).

Enhances understanding of the Milman-Schechtman theorem.

Abstract

For a symmetric convex body $K\subset\mathbb{R}^n$, the Dvoretzky dimension $k(K)$ is the largest dimension for which a random central section of $K$ is almost spherical. A Dvoretzky-type theorem proved by V.~D.~Milman in 1971 provides a lower bound for $k(K)$ in terms of the average $M(K)$ and the maximum $b(K)$ of the norm generated by $K$ over the Euclidean unit sphere. Later, V.~D.~Milman and G. Schechtman obtained a matching upper bound for $k(K)$ in the case when $\frac{M(K)}{b(K)}>c(\frac{\log(n)}{n})^{\frac{1}{2}}$. In this paper, we will give an elementary proof of the upper bound in Milman-Schechtman theorem which does not require any restriction on $M(K)$ and $b(K)$.