Upper Bound for the Dvoretzky Dimension in 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).