# 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} }

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…

## One Citation

Online learning with exponential weights in metric spaces

- Mathematics, Computer ScienceArXiv
- 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.

## References

SHOWING 1-10 OF 27 REFERENCES

On the geometry of metric measure spaces

- Mathematics
- 2006

AbstractWe introduce and analyze lower (Ricci) curvature bounds
$
\underline{{Curv}} {\left( {M,d,m} \right)}
$ ⩾ K for metric measure spaces
$
{\left( {M,d,m} \right)}
$. Our definition is based on…

Jensen’s inequality on convex spaces

- Mathematics
- 2014

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,…

On the geometry of metric measure spaces. II

- Mathematics
- 2006

AbstractWe introduce a curvature-dimension condition CD (K, N) for metric measure spaces. It is more restrictive than the curvature bound
$\underline{{{\text{Curv}}}} {\left( {M,{\text{d}},m}…

Riemannian Lp center of mass: existence, uniqueness, and convexity

- Mathematics
- 2011

Let be a complete Riemannian manifold and a probability measure on . Assume . We derive a new bound (in terms of , the injectivity radius of and an upper bound on the sectional curvatures of ) on the…

Heat Kernel Comparison on Alexandrov Spaces with Curvature Bounded Below

- Mathematics
- 2004

In this paper the comparison result for the heat kernel on Riemannian manifolds with lower Ricci curvature bound by Cheeger and Yau (1981) is extended to locally compact path metric spaces (X,d) with…

Probability Measures on Metric Spaces of Nonpositive Curvature

- Mathematics
- 2003

We present an introduction to metric spaces of nonpositive curvature (”NPC spaces”) and a discussion of barycenters of probability measures on such spaces. In our introduction to NPC spaces, we will…

Convergence rates for empirical barycenters in metric spaces: curvature, convexity and extendable geodesics

- MathematicsProbability Theory and Related Fields
- 2019

This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption…

Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence

- Mathematics
- 1990

This paper uses probability theory to provide an approach to the Dirichlet problem for harmonic maps. The probabilistic tools used are those of manifold-valued Brownian motion and Γ-martingales.…

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

- Mathematics
- 2005

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence…