Equality characterization and stability for entropy inequalities ∗ Elisabeth M. Werner † and T¨ urkay Yolcu December 18, 2013 Abstract We characterize the equality case in a recently established en- tropy inequality. To do so, we show that characterization of equal- ity is equivalent to uniqueness of the solution of a certain Monge Amp` ere differential equation. We prove the uniqueness of the so- lution 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 con- vex 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. 1 Introduction Recall that a measure μ with density e −ψ with respect to the Lebesgue measure is called log-concave if ψ is a convex function. For such log- concave probability measures the following reverse log Sobolev inequal- ity was established recently [3] (see also [14]),  log ( det(∇ 2 ψ) ) dμ ≤ 2  S (γ n ) − S (μ)  . (1) ∗ Keywords: entropy, divergence, affine isoperimetric inequalities, log Sobolev in- equalities, Monge Amp` ere, optimal mass transport. 2010 Mathematics Subject Clas- sification: 46B, 52A20, 60B, 35J † Partially supported by an NSF grant 1