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

TL;DR

This paper explores the geometric and metric properties of Fréchet means on the real line and extends these concepts to higher dimensions using Hessian Riemannian geometry, introducing new interpretations and dualities.

Contribution

It generalizes the interpretation of Fréchet means as centers of mass and introduces dual coordinate systems and quasi-arithmetic means in Hessian Riemannian manifolds.

Findings

Interior points of open intervals can be viewed as Fréchet means with specific metrics

Dual Fréchet/Karcher means are related by convex duality in dual coordinates

Squared Hessian metrics correspond to Euclidean geometry with Bregman centroids

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.