Journal of Convex Analysis 27 (2020), No. 4, 1195--1218
Copyright Heldermann Verlag 2020
Convex Envelopes on Trees
Leandro M. Del Pezzo
CONICET and Dep. de Matemática, FCEyN, Universidad de Buenos Aires, Argentina
ldpezzo@dm.uba.ar
Nicolas Frevenza
Dep. de Métodos Cuantitativos, FCEA, Universidad de la República, Montevideo, Uruguay
nfrevenza@dm.uba.ar
Julio D. Rossi
CONICET and Dep. de Matemática, FCEyN, Universidad de Buenos Aires, Argentina
jrossi@dm.uba.ar
We introduce two notions of convexity for an infinite regular tree. For these two notions we show that given a continuous boundary datum there exists a unique convex envelope on the tree and characterize the equation that this envelope satisfies. We also relate the equation with two versions of the Laplacian on the tree. Moreover, for a function defined on the tree, the convex envelope turns out to be the solution to the obstacle problem for this equation.
Keywords: Convexity on graphs, Laplacian on graphs, convex envelopes.
MSC: 35R02, 05C05, 52A41.
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