Geodesic
Convex Envelopes on Trees
Journal of convex analysis, Tome 27 (2020) no. 4, pp. 1195-1218
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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.
Classification : 35R02, 05C05, 52A41
Mots-clés : Convexity on graphs, Laplacian on graphs, convex envelopes 
@article{JCA_2020_27_4_JCA_2020_27_4_a6,
     author = {L. M. Del Pezzo and N. Frevenza and J. D. Rossi},
     title = {Convex {Envelopes} on {Trees}},
     journal = {Journal of convex analysis},
     pages = {1195--1218},
     year = {2020},
     volume = {27},
     number = {4},
     url = {http://geodesic.mathdoc.fr/item/JCA_2020_27_4_JCA_2020_27_4_a6/}
}
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L. M. Del Pezzo; N. Frevenza; J. D. Rossi. Convex Envelopes on Trees. Journal of convex analysis, Tome 27 (2020) no. 4, pp. 1195-1218. http://geodesic.mathdoc.fr/item/JCA_2020_27_4_JCA_2020_27_4_a6/