Jensen's inequality in geodesic spaces with lower bounded curvature

TL;DR

This paper extends Jensen's inequality to geodesic spaces with lower bounded curvature, establishing conditions under which the inequality holds for geodesically convex functions with respect to barycenters of probability measures.

Contribution

It generalizes Jensen's inequality to Alexandrov spaces with curvature bounds, using tangent cone properties and semi-concave function gradients.

Findings

Jensen's inequality holds for geodesically convex functions in lower bounded curvature spaces.

The proof utilizes tangent cone analysis and semi-concave function gradients.

Conditions on the function and measure ensure the inequality's validity.

Abstract

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 $x^{*}\in M$. We show that for any geodesically $\alpha$-convex function $f:M\to \mathbb R$, for $\alpha\in \mathbb R$, the inequality \[f(x^*)\le \int_M (f -\frac{\alpha}{2}d^2(x^*,.))\,{\rm d}\mu,\] holds provided $f$ is locally Lipschitz at $x^*$ 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.