This is an announcement for the paper "Random version of Dvoretzky's theorem in $\ell_p^n$" by Grigoris Paouris, Petros Valettas and Joel Zinn. Abstract: We study the dependence on $\varepsilon$ in the critical dimension $k(n, p, \varepsilon)$ that one can find random sections of the $\ell_p^n$-ball which are $(1+\varepsilon)$-spherical. For any fixed $n$ we give lower estimates for $k(n, p, \varepsilon)$ for all eligible values $p$ and $\varepsilon$, which agree with the sharp estimates for the extreme values $p = 1$ and $p = \infty$. In order to do so, we provide bounds for the gaussian concentration of the $\ell_p$-norm. Archive classification: math.FA Mathematics Subject Classification: 46B06, 46B07, 46B09 Remarks: 45 pages Submitted from: valettasp@missouri.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1510.07284 or http://arXiv.org/abs/1510.07284