In this paper we prove a fractional version of a Caffarelli–Kohn–Nirenberg type interpolation inequality on hypersurfaces \(M\subset \mathbb {R}^{n+1}\) which are boundaries of convex sets. The inequality carries a universal constant independent of M and involves the fractional mean curvature of M. In particular, it interpolates between the fractional Micheal-Simon Sobolev inequality recently obtained by Cabré, Cozzi, and the first author, and a new fractional Hardy inequality on M. Our method, when restricted to the plane case \(M=\mathbb {R}^n\) , gives a new simple proof of the fractional Hardy inequality. To obtain the fractional Hardy inequality on a hypersurface, we establish an inequality which bounds a weighted perimeter of M by the standard perimeter of M (modulo a universal constant), and which is valid for all convex hypersurfaces M.
