On the equality conditions of the Brunn-Minkowski theorem
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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 $\lambda \in [0,1]$, \[ V_{n}((1-\lambda )K + \lambda L)^{1/n} \geq (1-\lambda ) V_{n}(K)^{1/n} + \lambda V_{n}(L)^{1/n}.\] 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.References
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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
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 3719-3726
- MSC (2010): Primary 52A20, 52A38, 52A39, 52A40
- DOI: https://doi.org/10.1090/S0002-9939-2011-10822-0
- MathSciNet review: 2813401