Jensen’s inequality on convex spaces

Abstract

We prove a Jensen’s inequality on \(p\)-uniformly convex space in terms of \(p\)-barycenters of probability measures with \((p-1)\)-th moment with \(p\in ]1,\infty [\) under a geometric condition, which extends the results in Kuwae (Jensen’s inequality over CAT\((\kappa )\)-space with small diameter. In: Proceedings of Potential Theory and Stochastics, Albac Romania, pp. 173–182. Theta Series in Advanced Mathematics, vol. 14. Theta, Bucharest, 2009) , Eells and Fuglede (Harmonic maps between Riemannian polyhedra. In: Cambridge Tracts in Mathematics, vol. 142. Cambridge University Press, Cambridge, 2001) and Sturm (Probability measures on metric spaces of nonpositive curvature. Probability measures on metric spaces of nonpositive curvature. In: Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 357–390. Contemporary Mathematics, vol. 338. American Mathematical Society, Providence, 2003). As an application, we give a Liouville’s theorem for harmonic maps described by Markov chains into \(2\)-uniformly convex space satisfying such a geometric condition. An alternative proof of the Jensen’s inequality over Banach spaces is also presented.