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

Here we examine some connections between the notions of generalized arithmetic means, geodesics, Lagrange-Hamilton dynamics and Bregman divergences. In a previous paper we developed a predictive interpretation of generalized arithmetic means. That work was more probabilistically oriented. Here we take a geometric turn, and see that generalized arithmetic means actually minimize a geodesic distance on $\mathbb{R}^n.$ Such metrics might result from pull-backs of the Euclidean metric in $\mathbb{R}^n.$ We shall furthermore see that in some cases these pull-backs might coincide with the Hessian of a convex function. This occurs when the Hessian of a convex function has 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 by the convex function and the geodesic distance in the metric defined by its Hessian.