The existence of continuations for different types of metrics

TL;DR

This paper generalizes criteria for extending partially defined metrics and ultrametrics using graph theory and triangle functions, providing explicit formulas for maximal continuations across various semimetric types.

Contribution

It introduces a unified approach to determine the existence of continuations for diverse semimetrics using triangle functions, extending previous results.

Findings

Criteria for continuation of metrics and ultrametrics are generalized.

Explicit formulas for maximal continuations are derived.

Applicable to a broad class of semimetrics, including multiplicative and power triangle inequalities.

Abstract

The problems of continuation of a partially defined metric and a partially defined ultrametric were considered in (O. Dovgoshey, O. Martio and M. Vuorinen, Metrization of weighted graphs, Ann. Comb., 17:455--476, 2013) and (A. A. Dovgoshey and E. A. Petrov, Subdominant pseudoultrametric on graphs, Sb. Math., 204:1131--1151, 2013), respectively. Using the language of graph theory we generalize the criteria of existence of continuation obtained in these papers. For these purposes we use the concept of a triangle function introduced by M. Bessenyei and Z. P\'ales in (M. Bessenyei and Z. P\'ales, A contraction principle in semimetric spaces, J. Nonlinear Convex Anal., 18:515--524, 2017), which gives a generalization of the triangle inequality in metric spaces. The obtained result allows us to get criteria of the existence of continuation for a wide class of semimetrics including not only metrics and ultrametrics, but also multiplicative metrics and semimetrics with power triangle inequality. Moreover, the explicit formula for the maximal continuations is also obtained.