A von Neumann-Jordan Constant of Non-Normable Metrics

TL;DR

This paper extends the concept of the von Neumann-Jordan constant to non-normable metrics on vector spaces, providing conditions, examples, and formulas for specific metrics, thus broadening its applicability beyond normed spaces.

Contribution

It introduces a framework for analyzing the von Neumann-Jordan constant in non-normable metric spaces, including formulas for p-metrics and conditions for existing results to hold.

Findings

Established conditions under which norm results apply to non-normable metrics

Derived formulas for the von Neumann-Jordan constant of p-metrics

Provided examples and counterexamples to illustrate the theory

Abstract

The paper studies a generalized von Neumann-Jordan constant of non-normable metrics on vector spaces. To the best of our knowledge, all existing results of the von Neumann-Jordan constant and its generalizations have been established only in the normed setting. We identify reasonable conditions on non-normable metrics under which results known for norms remain valid. Several examples and counterexamples are provided to justify the established results. The computation for a class of non-normable metrics on product spaces is also investigated. In particular, we give precise formulas for the generalized von Neumann-Jordan constant of p-metrics under a metric-type Clarkson inequality. Comparisons with existing results are discussed throughout the paper whenever applicable.