  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.
