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 ≤ p < ∞. Here we investigate the asymptotic behavior of the planar p-flow for p = ∞ in the class of smooth, origin-symmetric convex bodies. First, we prove that the ∞-flow evolves suitably normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo SL(2). Second, using the ∞-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 C ∞ topology.2010 Mathematics Subject Classification. Primary 52A40, 53C44, 52A10; Secondary 35K55, 53A15.