This is an announcement for the paper "Equality characterization and stability for entropy inequalities" by Elisabeth M. Werner and Turkay Yolcu. Abstract: We characterize the equality case in a recently established entropy inequality. To do so, we show that characterization of equality is equivalent to uniqueness of the solution of a certain Monge Ampere differential equation. We prove the uniqueness of the solution using methods from mass transport, due to Brenier, and Gangbo-McCann. We then give stability versions for this entropy inequality, as well as for a reverse log Sobolev inequality and for the L_p-affine isoperimetric inequalities for both, log concave functions and convex bodies. In the case of convex bodies such stability results have only been known in all dimensions for p=1 and for p > 1 only for 0-symmetric bodies in the plane. Archive classification: math.FA Submitted from: elisabeth.werner@case.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1312.4148 or http://arXiv.org/abs/1312.4148