  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.
