The planar Busemann-Petty centroid inequality and its stability
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Abstract:
In Centro-affine invariants for smooth convex bodies [Int. Math. Res. Notices. DOI 10.1093/imrn/rnr110, 2012] 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 appropriately 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.References
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Additional Information
- Mohammad N. Ivaki
- Affiliation: Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 0, 1040 Wien, Austria
- Email: mohammad.ivaki@tuwien.ac.at
- Received by editor(s): April 15, 2014
- Published electronically: September 9, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 3539-3563
- MSC (2010): Primary 52A40, 53C44, 52A10; Secondary 35K55, 53A15
- DOI: https://doi.org/10.1090/tran/6503
- MathSciNet review: 3451885