Convex and quasiconvex functions in metric graphs

Abstract

<p style='text-indent:20px;'>We study convex and quasiconvex functions on a metric graph. Given a set of points in the metric graph, we consider the largest convex function below the prescribed datum. We characterize this largest convex function as the unique largest viscosity subsolution to a simple differential equation, <inline-formula><tex-math id="M1">\begin{document}$ u'' = 0 $\end{document}</tex-math></inline-formula> on the edges, plus nonlinear transmission conditions at the vertices. We also study the analogous problem for quasiconvex functions and obtain a characterization of the largest quasiconvex function that is below a given datum.</p>