Quasiconvex functions on regular trees

Leandro M. Del Pezzo; Nicolas Frevenza; Julio D. Rossi

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arXiv · 2006.11568

Quasiconvex functions on regular trees

Leandro M. Del Pezzo·Nicolas Frevenza·Julio D. Rossi

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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.

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Leandro M. Del Pezzo, Nicolas Frevenza, Julio D. Rossi. 2020-06-20. Quasiconvex functions on regular trees. https://arxiv.org/abs/2006.11568

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