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May 2016 Dar’s conjecture and the log–Brunn–Minkowski inequality
Dongmeng Xi, Gangsong Leng
J. Differential Geom. 103(1): 145-189 (May 2016). DOI: 10.4310/jdg/1460463565

Abstract

In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar $o$-symmetric convex bodies. In this paper, we give a positive answer to Dar’s conjecture for all planar convex bodies. We also give the equality condition of this stronger inequality.

For planar $o$-symmetric convex bodies, the log–Brunn–Minkowski inequality was established by Böröczky, Lutwak, Yang, and Zhang in 2012. It is stronger than the classical Brunn–Minkowski inequality, for planar $o$-symmetric convex bodies. Gaoyong Zhang asked if there is a general version of this inequality. Fortunately, the solution of Dar’s conjecture, especially, the definition of “dilation position”, inspires us to obtain a general version of the log–Brunn–Minkowski inequality. As expected, this inequality implies the classical Brunn–Minkowski inequality for all planar convex bodies.

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Dongmeng Xi. Gangsong Leng. "Dar’s conjecture and the log–Brunn–Minkowski inequality." J. Differential Geom. 103 (1) 145 - 189, May 2016. https://doi.org/10.4310/jdg/1460463565

Information

Published: May 2016
First available in Project Euclid: 12 April 2016

zbMATH: 1348.52006
MathSciNet: MR3488132
Digital Object Identifier: 10.4310/jdg/1460463565

Rights: Copyright © 2016 Lehigh University

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Vol.103 • No. 1 • May 2016
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