On the stability of the $p$-affine isoperimetric inequality

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\ge 1$ 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.