Metric compatibility and determination in complete metric spaces

TL;DR

This paper extends metric determination results to bounded functions in complete metric spaces by introducing an asymptotic control assumption, removing the need for compactness, and applies to various slope concepts.

Contribution

It generalizes previous results by removing the compactness requirement and establishing determination for bounded functions using an asymptotic behavior assumption.

Findings

Determination results hold for bounded functions with asymptotic control.

Applicable to local slope, average descent operators, and global slope.

Extends previous metric determination theorems to broader function classes.

Abstract

It was established in [8] that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if $f$ is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of [8] and [18].