Quasiconvex functions on regular trees

TL;DR

This paper defines quasiconvex functions on infinite regular trees, characterizes their envelopes via a median-based mean value property, and relates them to obstacle problems, expanding the understanding of convexity in tree structures.

Contribution

It introduces a novel definition of quasiconvexity on trees, proves existence and uniqueness of envelopes, and characterizes the associated mean value equation.

Findings

Existence and uniqueness of quasiconvex envelopes for continuous boundary data

Characterization of the envelope via a median-based mean value property

Connection between quasiconvex envelopes and obstacle problems

Abstract

We introduce a definition of a quasiconvex function on an infinite directed regular tree that depends on what we understood by a segment on the tree. Our definition is based on thinking on segments as sub-trees with the root as the midpoint of the segment. A convex set in the tree is then a subset such that it contains every midpoint of every segment with terminal nodes in the set. Then a quasiconvex function is a real map on the tree such that every level set is a convex set. For this concept of quasiconvex functions on a tree, we show that given a continuous boundary datum there exists a unique quasiconvex envelope on the tree and we characterize the equation that this envelope satisfies. It turns out that this equation is a mean value property that involves a median among values of the function on successors of a given vertex. We also relate the quasiconvex envelope of a function defined inside the tree with the solution of an obstacle problem for this characteristic equation.