This is an announcement for the paper “Upper bound for the Dvoretzky dimension in Milman-Schechtman theorem” by Han Huanghttps://arxiv.org/find/math/1/au:+Huang_H/0/1/0/all/0/1, Feng Weihttps://arxiv.org/find/math/1/au:+Wei_F/0/1/0/all/0/1.
Abstract: For a symmetric convex body $K\subset R_n$, 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 $M(K)b(K)$>c(\log (n)/n)^{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 $b(K)$
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1612.03572