2023/05/26 by Wolf‐Jürgen Beyn, Beyn, Wolf-Jürgen
Physics and Astronomy · #15A99 #51F99 #Advanced Differential Geometry Research #FOS: Mathematics #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2305.17081
openalex publication_date 2023/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 points. We present two principles of constructing pseudo n-metrics. The first one uses the Vandermonde determinant while the second one uses exterior products and is related to the volume of the simplex spanned by the given points. We show that the second class of examples induces pseudo n-metrics on the unit sphere of a Hilbert space and on matrix manifolds such as the Stiefel and 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.