On a generalized notion of metrics

In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n -metric which assigns a value to a tuple of n ≥ 2 points. Some elementary properties of pseudo n -metrics are provided and their construction via exterior products is investigated. We d...

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Bibliographic Details
Published in:Aequationes mathematicae 2024-08, Vol.98 (4), p.953-977
Main Author: Beyn, Wolf-Jürgen
Format: Article
Language:English
Subjects:
Analysis
Combinatorics
Euclidean geometry
Manifolds (mathematics)
Mathematics
Mathematics and Statistics
Metric space
Vector spaces
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Summary:In these notes we generalize the notion of a (pseudo) metric measuring the distance of two points, to a (pseudo) n -metric which assigns a value to a tuple of n ≥ 2 points. Some elementary properties of pseudo n -metrics are provided and their construction via exterior products is investigated. We discuss some examples from the geometry of Euclidean vector spaces leading to pseudo n -metrics on the unit sphere, on the Stiefel manifold, and on the Grassmann manifold. Further, we construct a pseudo n -metric on hypergraphs and discuss the problem of generalizing the Hausdorff metric for closed sets to a pseudo n -metric.
ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-024-01092-y