Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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

Keywords

  • Convexity on graphs
  • Laplacian on graphs
  • convex envelopes

Mathematics Subject Classification

35R02, 05C05, 52A41

Suggested citation

L. M. Del Pezzo, N. Frevenza, J. D. Rossi. “Convex Envelopes on Trees.” Journal of Convex Analysis 27 (2020), No. 4, 1195–1218.

Copyright Banach Press 2020