The Heintze-Karcher inequality for metric measure spaces

In this note we prove the Heintze-Karcher inequality in the context of essentially non-branching metric measure spaces satisfying a lower Ricci curvature bound in the sense of Lott-Sturm-Villani. The proof is based on the the needle decomposition technique for metric measure spaces introduced by Cavalletti-Mondino. Moreover, in the class of spaces satisfying a Riemannian curvature-dimension condition with positive curvature the equality case is characterized.