This is an announcement for the paper "Functional inequalities derived from the Brunn-Minkowski inequalities for quermassintegrals" by Andrea Colesanti and Eugenia Saorin-Gomez. Abstract: We use Brunn-Minkowski inequalities for quermassintegrals to deduce a family of inequalities of Poincar\'e type on the unit sphere and on the boundary of smooth convex bodies in the $n$-dimensional Euclidean space. Archive classification: math.FA math.MG Mathematics Subject Classification: 52A20; 26D10 Remarks: 15 pages The source file(s), cs3.tex: 37802 bytes, is(are) stored in gzipped form as 0804.3867.gz with size 12kb. The corresponding postcript file has gzipped size 109kb. Submitted from: colesant@math.unifi.it The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0804.3867 or http://arXiv.org/abs/0804.3867 or by email in unzipped form by transmitting an empty message with subject line uget 0804.3867 or in gzipped form by using subject line get 0804.3867 to: math@arXiv.org.