Fractional convexity
Abstract
We introduce a notion of fractional convexity that extends naturally the usual notion of convexity in the Euclidean space to a fractional setting. With this notion of fractional convexity, we study the fractional convex envelope inside a domain of an exterior datum (the largest possible fractional convex function inside the domain that is below the datum outside) and show that the fractional convex envelope is characterized as a viscosity solution to a non-local equation that is given by the infimum among all possible directions of the dimensional fractional Laplacian. For this equation we prove existence, uniqueness and a comparison principle (in the framework of viscosity solutions). In addition, we find that solutions to the equation for the convex envelope are related to solutions to the fractional Monge-Ampere equation.
Cite
@article{arxiv.2009.04141,
title = {Fractional convexity},
author = {Leandro M. Del Pezzo and Alexander Quaas and Julio D. Rossi},
journal= {arXiv preprint arXiv:2009.04141},
year = {2021}
}
Comments
25 pages