  The anisotropic fractional $s$-perimeter with respect to a convex body $K$ in
$\rn$ is shown to converge to the anisotropic perimeter with respect to the
moment body of $K$ as $s\to 1^-$. A corresponding result is established for
anisotropic fractional $s$-seminorms on $BV(\rn)$ (generalizing results of
Bourgain, Brezis & Mironescu and D\'avila). The minimizers of the anisotropic
fractional $s$-isoperimetric inequality with respect to $K$ are shown to
converge to the moment body of $K$. Anisotropic fractional Sobolev inequalities
are established.
