  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.
