*Preprint*

**Inserted:** 22 jan 2020

**Last Updated:** 10 feb 2020

**Year:** 2020

**Notes:**

A big gap in the previous version is filled.

**Abstract:**

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Emery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincare inequality in the setting of ${\rm RCD}(K, \infty)$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce, Morgan etc.. Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of $\Phi$-entropy inequalities, the rigidity of the 1-Bakry-Emery inequality are of independent interest even in the smooth setting.

**Keywords:**
Ricci curvature, metric measure space, Bakry-Emery theory

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