Centro-Affine Invariants for Smooth Convex Bodies

Abstract

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.

Affine images of isotropic measures

“…In recent years, cone-volume measures have appeared in, e.g., [43,45,62,63,66,73]. Firey's Question asks if a body whose cone-volume measure is proportional to spherical Lebesgue measure on S n−1 must be a ball.…”

The planar Busemann-Petty centroid inequality and its stability

“…Employing the evolution equation of polar bodies was first introduced by Stancu [74] in the context of centro-affine normal flows. In [41,44] it was shown that the evolution equation of polar bodies combined with Tso's trick and SalkowskiKaltenbach-Hug identity (see [37, Theorem 2.8]) provide a useful tool for obtaining regularity of solutions to a class of geometric flows.…”

“…The flow whose definition will be given, was first introduced by Stancu in [74]. In [74] Stancu introduces new equicentro-affine invariants, and she provides a geometric interpretation of the L Φ affine surface area introduced by Ludwig and Reitzner [50].…”

“…The short time existence and uniqueness of solutions for smooth and strictly convex initial bodies follow from the strict parabolicity of the equation and were established in [74].…”

Some Affine Invariants Revisited

“…The present paper spun as a follow-up of [32] in which we introduced new SL(n)-invariants for smooth convex bodies. We started by searching for sharp affine inequalities satisfied by one such invariant derived, in a certain sense, from the centro-affine surface area.…”