Harmonic Center of an n-Simplex: Some Properties

TL;DR

This paper investigates properties of the harmonic center in bounded n-simplices, deriving conditions for boundedness, relationships between residuals at the harmonic center, and an invariant linear expression within the simplex.

Contribution

It introduces new conditions for boundedness of simplexes, explores residual relationships at the harmonic center, and develops an invariant linear expression within the simplex.

Findings

Derived a condition relating the rows of A for boundedness

Established a relationship between residuals at the harmonic center

Developed a linear invariant expression inside the simplex

Abstract

A simplex in n dimensions is defined by the usual (n+1) linear inequality constraints in n dimensions. Here we consider simplexes which are bounded sets. The harmonic center has been defined earlier for polytopes in general. A relationship between the rows of A is derived here, which must be satisfied for the simplex to be bounded. Using this condition, an interesting relationship is derived between the (n+1) residuals at the harmonic center of the simplex. Finally, a linear expression is developed which is invariant everywhere in the simplex.