  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.
