This is an announcement for the paper "The planar Busemann-Petty centroid inequality and its stability" by Mohammad N. Ivaki.

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 $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$, in the class of smooth, origin-symmetric convex bodies. The motivation is the Busemann-Petty centroid inequality. First, we prove that the $\infty$-flow evolves appropriately normalized origin-symmetric solutions to the unit disk in the Hausdorff metric, modulo $SL(2).$ Second, as an application of this weak convergence, we prove the planar Busemann-Petty centroid inequality in the of class convex bodies having the origin of the plane in their interiors. Third, using the $\infty$-flow, we prove a stability version of the planar Busemann-Petty centroid inequality, in the Banach-Mazur distance, in the class of origin-symmetric convex bodies. Fourth, we prove that the convergence in the Hausdorff metric can be improved to convergence in the $\mathcal{C}^{\infty}$ topology.

Archive classification: math.DG math.FA

Mathematics Subject Classification: Primary 52A40, 53C44, 52A10, Secondary 35K55, 53A15

Remarks: Two preprints unified into one

Submitted from: mohammad.ivaki@tuwien.ac.at

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1312.4834

or

http://arXiv.org/abs/1312.4834