On generalized metrics of Vandermonde type

TL;DR

This paper introduces a new class of generalized metrics based on the Vandermonde determinant, explores their geometric properties, extends the concept to higher dimensions, and applies it to point sets governed by differential equations.

Contribution

It proposes a novel Vandermonde-based generalized metric, analyzes its properties, and extends the concept to arbitrary linear spaces with applications to differential equations.

Findings

Vandermonde-based metrics satisfy generalized metric axioms.

Geometric consequences in the complex plane are established.

Application to point sets driven by linear ODEs demonstrates practical relevance.

Abstract

In a series of papers in the 1960's, S. G\"ahler defined and investigated so-called m-metric spaces and their topological properties. An m-metric assigns to any tuple of m+1 elements a real value (more generally an element in a partially odered set) which satisfies the generalized metric axioms of semidefiniteness, symmetry, and simplex inequality. In this contribution we consider a new type of generalized metric which is based on the Vandermonde determinant. We present some remarkable geometric consequences of the corresponding simplex inequality in the complex plane. Then we show that the Vandermonde principle of construction extends to linear spaces of arbitrary dimension by using symmetric multilinear maps of degree m(m + 1)/2. In particular, we analyze when this generalized metrics has the stronger property of definiteness. Finally, an application is provided to the m-metric of point sets when driven by the same linear ordinary differential equation.