  We extend the affine inequalities on $\mathbb{R}^n$ for Sobolev functions in
$W^{s,p}$ with $1 \leq p < n/s$ obtained recently by Haddad-Ludwig [16, 17] to
the remaining range $p \geq n/s$. For each value of $s$, our results are
stronger than affine Moser-Trudinger and Morrey inequalities. As a byproduct,
we establish the analog of the classical $L^p$ Bourgain-Brezis-Mironescu
inequalities related to the Moser-Trudinger case $p=n$. Our main tool is the
affine invariant provided by Besov polar projection bodies.
