Scales of Fr\'echet means and Karcher quasi-arithmetic means

Abstract

In this paper, we first prove that any interior point of an open interval of the real line can be interpreted as Fr\'echet means with respect to corresponding metric distances, thus extending the result of [Dinh et al., Mathematical Intelligencer 47.2 (2025)] which was restricted to intervals on the positive reals by using the family of power means: Our generic construction relies on the concept of scales of means that we demonstrate with the scale of exponential means and the scale of radical means. Second, we interpret those Fr\'echet means geometrically as the center of mass of any two distinct points on the Euclidean line expressed in various coordinate systems: Namely, by interpreting the Euclidean line as a 1D Hessian Riemannian manifold, we introduce pairs of dual Fr\'echet/Karcher means related by convex duality in dual coordinate systems. This result yields us to consider squared Hessian metrics in arbitrary dimension: We prove that these squared Hessian metrics amount to Euclidean geometry with the Riemannian center of mass expressed in primal coordinate systems as multivariate quasi-arithmetic means coinciding with left-sided Bregman centroids.