The limit as \(s\nearrow 1\) of the fractional convex envelope

Abstract

We study the behavior of the fractional convexity when the fractional parameter goes to 1. For any notion of convexity, the convex envelope of a datum prescribed on the boundary of a domain is defined as the largest possible convex function inside the domain that is below the datum on the boundary. Here we prove that the fractional convex envelope inside a strictly convex domain of a continuous and bounded exterior datum converges when \(s\nearrow 1\) to the classical convex envelope of the restriction to the boundary of the exterior datum.