This is an announcement for the paper "Interpolations, convexity and geometric inequalities" by Dario Cordero-Erausquin and Boaz Klartag. Abstract: We survey some interplays between spectral estimates of H\"ormander-type, degenerate Monge-Amp\`ere equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santal\'o or Busemann inequalities. Archive classification: math.FA math.CV Submitted from: cordero@math.jussieu.fr The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1109.3652 or http://arXiv.org/abs/1109.3652