Necessary and sufficient conditions for distances on the real line
Abstract
When dealing with certain mathematical problems, it is sometimes necessary to show that some function induces a metric on a certain space. When this function is not a well renowned example of a distance, one has to develop very particular arguments that appeal to the concrete expression of the function in order to do so. The main purpose of this paper is to provide several sufficient results ensuring that a function of two variables induces a distance on the real line, as well as some necessary conditions, together with several examples that show the applicability of these results. In particular, we show how a hypothesis about the sign of the cross partial derivative of the candidate to distance is helpful for deriving such kind of results.
Keywords
Cite
@article{arxiv.2305.13251,
title = {Necessary and sufficient conditions for distances on the real line},
author = {Daniel Cao Labora and Francisco Javier Fernández and Fernando Adrián F. Tojo and Carlos Villanueva},
journal= {arXiv preprint arXiv:2305.13251},
year = {2023}
}
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16 pages