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A LOCAL UNIQUENESS THEOREM FOR MINIMIZERS OF PETTY’S CONJECTURED PROJECTION INEQUALITY

Part of: Classical differential geometry General convexity

Published online by Cambridge University Press:  24 January 2018

Mohammad N. Ivaki
Mohammad N. Ivaki*
Affiliation:
Department of Mathematics, University of Toronto, Ontario, M5S 2E4, Canada email [email protected]
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Abstract

Employing the inverse function theorem on Banach spaces, we prove that in a $C^{2}(S^{n-1})$-neighborhood of the unit ball, the only solutions of $\unicode[STIX]{x1D6F1}^{2}K=cK$ are origin-centered ellipsoids. Here $K$ is an $n$-dimensional convex body, $\unicode[STIX]{x1D6F1}K$ is the projection body of $K$ and $\unicode[STIX]{x1D6F1}^{2}K=\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6F1}K).$

MSC classification

Primary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces) 53A15: Affine differential geometry
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 1 - 19
Copyright
Copyright © University College London 2018 

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