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Convex bodies with pinched Mahler volume under the centro-affine normal flows

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Abstract

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke–Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitably rescaled solutions converge sequentially to the unit ball in the \(\mathcal {C}^{\infty }\) topology modulo \(SL(n+1)\).

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Acknowledgments

I am indebted to the referees whose comments and suggestions have led to improvements of this article.

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Authors and Affiliations

  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10, 1040, Wien, Austria

    Mohammad N. Ivaki

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  1. Mohammad N. Ivaki
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Correspondence to Mohammad N. Ivaki.

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Communicated by Y. Giga.

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Ivaki, M.N. Convex bodies with pinched Mahler volume under the centro-affine normal flows. Calc. Var. 54, 831–846 (2015). https://doi.org/10.1007/s00526-014-0807-9

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