  In [Centro-affine invariants for smooth convex bodies, Int. Math. Res.
Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of
centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate
the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of
smooth, origin-symmetric convex bodies. First, we prove that the $\infty$-flow
evolves suitably normalized origin-symmetric solutions to the unit disk in the
Hausdorff metric, modulo $SL(2).$ Second, using the $\infty$-flow and a Harnack
estimate for this flow, we prove a stability version of the planar
Busemann-Petty centroid inequality in the Banach-Mazur distance. Third, we
prove that the convergence of normalized solutions in the Hausdorff metric can
be improved to convergence in the $\mathcal{C}^{\infty}$ topology.
