We present new Brascamp-Lieb and Poincaré type inequalities, for measures which are log-concave with respect to rotation invariant measures. Geometric applications for proving Brunn-Minkowski type inequalities will also be given. In particular, we will resolve the (B) conjecture and the Gardner-Zvavitch conjecture for rotation invariant log-concave measures. Based on joint work with Dario Cordero-Erausquin.
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