Convex and quasiconvex functions in metric graphs

TL;DR

This paper investigates convex and quasiconvex functions on metric graphs, characterizing the largest such functions below given data through differential equations and transmission conditions at vertices.

Contribution

It introduces a novel characterization of the largest convex and quasiconvex functions on metric graphs using viscosity solutions and differential equations.

Findings

Largest convex function characterized as viscosity subsolution of $u''=0$

Transmission conditions at vertices are crucial for the characterization

Extension of the theory to quasiconvex functions with similar properties

Abstract

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, $u''=0$ 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.