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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the equality conditions of the Brunn-Minkowski theorem


Author: Daniel A. Klain
Journal: Proc. Amer. Math. Soc. 139 (2011), 3719-3726
MSC (2010): Primary 52A20, 52A38, 52A39, 52A40
Published electronically: February 24, 2011
MathSciNet review: 2813401
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Abstract: This article describes a new proof of the equality condition for the Brunn-Minkowski inequality. The Brunn-Minkowski Theorem asserts that, for compact convex sets $ K,L \subseteq \mathbb{R}^n$, the $ n$-th root of the Euclidean volume $ V_n$ is concave with respect to Minkowski combinations; that is, for $ \lambda \in [0,1]$,

$\displaystyle V_{n}((1-\lambda)K + \lambda L)^{1/n} \geq (1-\lambda) V_{n}(K)^{1/n} + \lambda V_{n}(L)^{1/n}.$

The equality condition asserts that if $ K$ and $ L$ both have positive volume, then equality holds for some $ \lambda \in (0,1)$ if and only if $ K$ and $ L$ are homothetic.


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Additional Information

Daniel A. Klain
Affiliation: Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, Massachusetts 01854
Email: Daniel_Klain@uml.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10822-0
PII: S 0002-9939(2011)10822-0
Received by editor(s): May 9, 2010
Received by editor(s) in revised form: September 2, 2010
Published electronically: February 24, 2011
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.