This is an announcement for the paper "Small ball probability and Dvoretzky theorem" by Bo'az Klartag and Roman Vershynin. Abstract: Large deviation estimates are by now a standard tool inthe Asymptotic Convex Geometry, contrary to small deviationresults. In this note we present a novel application of a smalldeviations inequality to a problem related to the diameters of random sections of high dimensional convex bodies. Our results imply an unexpected distinction between the lower and the upper inclusions in Dvoretzky Theorem. Archive classification: Functional Analysis; Probability Mathematics Subject Classification: 46B07; 60F10 The source file(s), diameters.tex: 30564 bytes, is(are) stored in gzipped form as 0410001.gz with size 10kb. The corresponding postcript file has gzipped size 49kb. Submitted from: vershynin@math.ucdavis.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0410001 or http://arXiv.org/abs/math.FA/0410001 or by email in unzipped form by transmitting an empty message with subject line uget 0410001 or in gzipped form by using subject line get 0410001 to: math@arXiv.org.