  Employing the affine normal flow, we prove a stability version of the
$p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class
of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric
convex body in $\mathbb{R}^2$ such that it has area $\pi$ and its $p$-affine
perimeter is close enough to the one of an ellipse with the same area, then,
after applying a special linear transformation, $K$ is close to an ellipse in
the Hausdorff distance.
