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