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A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature

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Abstract

In the quadratic Aleksandrov-Fenchel inequality for mixed volumes, stated as inequality (1) below, whereC 1, ...,C n-2 are smooth convex bodies, equality holds only if the convex bodiesK andL are homothetic. Under stronger regularity assumptions onC 1,...,C n-2, a stability estimate is proved, expressing thatK andL are close to homothetic if equality is satisfied approximately. This is applied to estimate explicitly the deviation of a closed convex hypersurface with mean curvature close to one from a unit sphere.

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  1. Mathematisches Institut der Universität, Albertstr. 23 b, D-7800, Freiburg i. Br.

    Rolf Schneider

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  1. Rolf Schneider
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Schneider, R. A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature. Manuscripta Math 69, 291–300 (1990). https://doi.org/10.1007/BF02567927

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  • DOI: https://doi.org/10.1007/BF02567927

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