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We introduce a definition of a quasiconvex function on an infinite directedregular tree that depends on what we understood by a segment on the tree. Ourdefinition is based on thinking on segments as sub-trees with the root as themidpoint of the segment. A convex set in the tree is then a subset such that itcontains every midpoint of every segment with terminal nodes in the set. Then aquasiconvex function is a real map on the tree such that every level set is aconvex set. For this concept of quasiconvex functions on a tree, we show thatgiven a continuous boundary datum there exists a unique quasiconvex envelope onthe tree and we characterize the equation that this envelope satisfies. Itturns out that this equation is a mean value property that involves a medianamong values of the function on successors of a given vertex. We also relatethe quasiconvex envelope of a function defined inside the tree with thesolution of an obstacle problem for this characteristic equation.

Quasiconvex functions on regular trees