On fractional Hardy inequalities in convex sets

We prove a Hardy inequality on convex sets, for fractional Sobolev-Slobodeckiĭ spaces of order $(s, p)$. The proof is based on the fact that in a convex set the distance from the boundary is a superharmonic function, in a suitable sense. The result holds for every $1<p<∞$ and zhongwenzy $0<s<1$, with a constant which is stable as s goes to 1.