Year: 2025
Type: article
Abstract: In a series of papers in the 1960’s, S. Gähler defined and investigated so-called m -metric spaces and their topological properties. An m -metric assigns to any tuple of $$m+1$$ m + 1 elements a real value (more generally an element in a partially ordered 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 $$\frac{1}{2}m(m+1)$$ 1 2 m ( m + 1 ) . In particular, we analyze when this generalized metrics has the stronger property of definiteness, i.e. two of its arguments are equal if the metric vanishes. Finally, an application is provided to the m -metric of point sets when driven by the same linear ordinary differential equation.
Author: Wolf-Jürgen Beyn
Institution: Bielefeld University
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Funder: Universität Bielefeld
OpenAlex ID: W4409920690