Jensen's inequality in geodesic spaces with lower bounded curvature

Let $(M,d)$ be a separable and complete geodesic space with curvature lower bounded, by $\kappa \in \mathbb{R}$, in the sense of Alexandrov. Let $\mu$ be a Borel probability measure on $M$, such that $\mu \in {\mathcal{P}}_2(M)$, and that has at least one barycenter $x^{\ast }\in M$. We show that for any geodesically $\alpha$-convex function $f:M\to \mathbb{R}$, for $\alpha \in \mathbb{R}$, the inequality $$f(x^{\ast })\le {\int }_M(f-\frac{\alpha }{2}{d}^2(x^{\ast },\mathrm{.}))\mathrm{d}\mu ,$$ holds provided $f$ is locally Lipschitz at $x^{\ast }$ and either positive or in $L^1(\mu )$. Our proof relies on the properties of tangent cones at barycenters and on the existence of gradients for semi-concave functions in spaces with lower bounded curvature.