This is an announcement for the paper "Asymptotic Smoothness, Convex Envelopes and Polynomial Norms" by Jesus A. Jaramillo, Raquel Gonzalo and Diego Yanez. Abstract: We introduce a suitable notion of asymptotic smoothness on infinite dimensional Banach spaces, and we prove that, under some structural restrictions on the space, the convex envelope of an asymptotically smooth function is asymptotically smooth. Furthermore, we study convexity and smoothness properties of polynomial norms, and we obtain that a polynomial norm of degree N has modulus of convexity of power type N. Archive classification: math.FA Submitted from: jaramil@mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1512.05407 or http://arXiv.org/abs/1512.05407