Fractional convexity

TL;DR

This paper introduces a new concept of fractional convexity extending classical convexity to a fractional setting, characterizes the fractional convex envelope as a viscosity solution to a non-local equation, and explores its relation to fractional Monge-Ampère equations.

Contribution

It defines fractional convexity, studies the fractional convex envelope, and characterizes it via a non-local PDE, establishing existence, uniqueness, and connections to fractional Monge-Ampère equations.

Findings

Fractional convex envelope characterized as a viscosity solution.

Existence and uniqueness of solutions to the non-local equation.

Connections established between fractional convex envelope and fractional Monge-Ampère equation.

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 $1-$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.