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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The planar Busemann-Petty centroid inequality and its stability


Author: Mohammad N. Ivaki
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
Published electronically: September 9, 2015
MathSciNet review: 3451885
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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.


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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

DOI: https://doi.org/10.1090/tran/6503
Keywords: Affine support function, Banach-Mazur distance, centro-affine normal flow, centroid body, geometric evolution equation, Hausdorff metric, Busemann-Petty centroid inequality, stability
Received by editor(s): April 15, 2014
Published electronically: September 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society

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