Affine fractional Sobolev and isoperimetric inequalities

Sharp affine fractional Sobolev inequalities for functions on $\mathbb{R}^n$ are established. For each $0 \lt s \lt 1$ , the new inequalities are significantly stronger than (and directly imply) the sharp fractional Sobolev inequalities of Almgren and Lieb. In the limit as $s \rightarrow 1^{-}$ , the new inequalities imply the sharp affine Sobolev inequality of Gaoyong Zhang. As a consequence, fractional Petty projection inequalities are obtained which are stronger than the fractional Euclidean isoperimetric inequalities, and a natural conjecture for radial mean bodies is proved.