Geometric representation of the weighted harmonic mean of $n$ positive values and potential uses

TL;DR

This paper explores the geometric interpretation of the weighted harmonic mean of positive numbers and discusses its properties to develop numerical approximation methods for multidimensional discontinuities.

Contribution

It introduces a geometric framework for the weighted harmonic mean and establishes properties useful for multidimensional approximation methods.

Findings

Proves key properties of the weighted harmonic mean

Develops numerical approximation methods for discontinuities

Provides geometric interpretation of means

Abstract

This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of $n$ positive real numbers. Together with this interpretation, we prove some relevant properties that will allow us to define numerical approximation methods in several dimensions adapted to discontinuities.