  The classical Michael-Simon and Allard inequality is a Sobolev inequality for
functions defined on a submanifold of Euclidean space. It is governed by a
universal constant independent of the manifold, but displays on the right-hand
side an additional $L^p$ term weighted by the mean curvature of the underlying
manifold. We prove here a fractional version of this inequality on
hypersurfaces of Euclidean space that are boundaries of convex sets. It
involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm
weighted by the fractional mean curvature of the hypersurface.
  As an application, we establish a new upper bound for the maximal time of
existence in the smooth fractional mean curvature flow of a convex set. The
bound depends on the perimeter of the initial set instead of on its diameter.
