• Corpus ID: 226976097

# Jensen's inequality in geodesic spaces with lower bounded curvature

@article{Paris2020JensensII,
title={Jensen's inequality in geodesic spaces with lower bounded curvature},
author={Quentin Paris},
journal={arXiv: Metric Geometry},
year={2020}
}
• Q. Paris
• Published 17 November 2020
• Mathematics
• arXiv: Metric Geometry
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…
1 Citations
Online learning with exponential weights in metric spaces
• Q. Paris
• Mathematics, Computer Science
ArXiv
• 2021
This paper extends the analysis of the exponentially weighted average forecaster, traditionally studied in a Euclidean settings, to a more abstract framework using the notion of barycenters, a suitable version of Jensen’s inequality and a synthetic notion of lower curvature bound in metric spaces known as the measure contraction property.

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