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Here we examine some connections between the notions of generalizedarithmetic means, geodesics, Lagrange-Hamilton dynamics and Bregmandivergences. In a previous paper we developed a predictive interpretation ofgeneralized arithmetic means. That work was more probabilistically oriented.Here we take a geometric turn, and see that generalized arithmetic meansactually minimize a geodesic distance on $\mathbb{R}^n.$ Such metrics mightresult from pull-backs of the Euclidean metric in $\mathbb{R}^n.$ We shallfurthermore see that in some cases these pull-backs might coincide with theHessian of a convex function. This occurs when the Hessian of a convex functionhas a square root that is the Jacobian of a diffeomorphism in $\mathbb{R}^n.$In this case we obtain a comparison between the Bregman divergence defined bythe convex function and the geodesic distance in the metric defined by itsHessian.

From generalized arithmetic means to geodesics to Hamilton dynamics to Bregman divergences