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Bulletin of the American Mathematical Society

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The Brunn-Minkowski inequality

Author: R. J. Gardner
Journal: Bull. Amer. Math. Soc. 39 (2002), 355-405
MSC (2000): Primary 26D15, 52A40
Published electronically: April 8, 2002
MathSciNet review: 1898210
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Abstract: In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of ${\mathbb R}^n$, and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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R. J. Gardner
Affiliation: Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063

Keywords: Brunn-Minkowski inequality, Minkowski's first inequality, Pr\'{e}kopa-Leindler inequality, Young's inequality, Brascamp-Lieb inequality, Barthe's inequality, isoperimetric inequality, Sobolev inequality, entropy power inequality, covariogram, Anderson's theorem, concave function, concave measure, convex body, mixed volume
Received by editor(s): February 1, 2001
Received by editor(s) in revised form: November 28, 2001
Published electronically: April 8, 2002
Additional Notes: Supported in part by NSF Grant DMS 9802388.
Article copyright: © Copyright 2002 American Mathematical Society