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Any interior point of a finite interval on the real line can be interpreted as dual Fréchet means

Frank Nielsen

TL;DR

The paper addresses how any interior point $c$ of a finite interval can be realized as a midpoint with respect to suitably chosen distances. It develops a general theorem: for any scale of strictly monotone generators $\{s_\alpha\}$ yielding a scale of quasi-arithmetic means $m_{s_\alpha}$, the distance $d_{s_\alpha}(x,y)=|s_\alpha(x)-s_\alpha(y)| makes every $c\in(a,b)$ a midpoint, broadening the Fréchet-mean viewpoint beyond power means. It provides concrete realizations via two families—exponential and radical means—demonstrating how midpoints sweep the entire interval and connecting these to geometric interpretations in coordinate charts and to a dual-scale framework from convex analysis. The dual-scales perspective interprets the 1D Euclidean line as a Hessian manifold, linking primal and dual mean generators through Legendre duality, and highlighting a new convex-duality lens on means and midpoints with potential broader implications for generalized center-of-m.mass representations.

Abstract

In this note, we show that any interior point of an open interval on the real line can be interpreted as a pair of dual Fréchet quasi-arithmetic means associated with dual metric distances. A Riemannian interpretation of the Euclidean line brings to light a novel concept of dual scales of means, grounded in convex duality and of independent interest.

Any interior point of a finite interval on the real line can be interpreted as dual Fréchet means

TL;DR

The paper addresses how any interior point of a finite interval can be realized as a midpoint with respect to suitably chosen distances. It develops a general theorem: for any scale of strictly monotone generators yielding a scale of quasi-arithmetic means , the distance c\in(a,b)$ a midpoint, broadening the Fréchet-mean viewpoint beyond power means. It provides concrete realizations via two families—exponential and radical means—demonstrating how midpoints sweep the entire interval and connecting these to geometric interpretations in coordinate charts and to a dual-scale framework from convex analysis. The dual-scales perspective interprets the 1D Euclidean line as a Hessian manifold, linking primal and dual mean generators through Legendre duality, and highlighting a new convex-duality lens on means and midpoints with potential broader implications for generalized center-of-m.mass representations.

Abstract

In this note, we show that any interior point of an open interval on the real line can be interpreted as a pair of dual Fréchet quasi-arithmetic means associated with dual metric distances. A Riemannian interpretation of the Euclidean line brings to light a novel concept of dual scales of means, grounded in convex duality and of independent interest.

Paper Structure

This paper contains 6 sections, 3 theorems, 23 equations.

Table of Contents

  1. Introduction: Distances, Fréchet means, and quasi-arithmetic means
  2. Midpoints from scales of means
  3. Example 1: Scale of exponential means on $I=\mathbb{R}$
  4. Example 2: Scale of radical means on $\mathbb{R}_+$
  5. A geometric interpretation: Center of mass of two points expressed in various coordinate charts
  6. Dual scales of means from convex analysis

Key Result

Theorem 1

Let $\{s_\alpha\}_{\alpha\in\mathbb{R}}$ be a family of strictly monotone and differentiable functions yielding a scale $\{m_{s_\alpha}\}$ of quasi-arithmetic means, where $m_{s_\alpha}: I\times I\rightarrow I$ for $I\subset\mathbb{R}$. Then any scalar $c$ of a given interval $(a,b)\subset I$ is the

Theorems & Definitions (7)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Example 1