15 March 2022 The extremals of Minkowski’s quadratic inequality
Yair Shenfeld, Ramon van Handel
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Duke Math. J. 171(4): 957-1027 (15 March 2022). DOI: 10.1215/00127094-2021-0033


In a seminal paper “Volumen und Oberfläche” (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to date only been resolved in special cases. In this paper, we completely settle the extremals of Minkowski’s quadratic inequality, confirming a conjecture of R. Schneider. Our proof is based on the representation of mixed volumes of arbitrary convex bodies as Dirichlet forms associated to certain highly degenerate elliptic operators. A key ingredient of the proof is a quantitative rigidity property associated to these operators.


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Yair Shenfeld. Ramon van Handel. "The extremals of Minkowski’s quadratic inequality." Duke Math. J. 171 (4) 957 - 1027, 15 March 2022. https://doi.org/10.1215/00127094-2021-0033


Received: 26 July 2019; Revised: 8 April 2021; Published: 15 March 2022
First available in Project Euclid: 14 March 2022

MathSciNet: MR4393790
zbMATH: 1487.52013
Digital Object Identifier: 10.1215/00127094-2021-0033

Primary: 52A39
Secondary: 52A40 , 58J50

Keywords: Alexandrov–Fenchel inequality , convex geometry , extremum problems , Minkowski’s quadratic inequality , mixed volumes

Rights: Copyright © 2022 Duke University Press


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Vol.171 • No. 4 • 15 March 2022
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