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Stein and Shakarchi, in their book Real Analysis, the third volume of the Princeton Lectures in Analysis series, give a proof of the isoperimetric inequality for closed rectifiable curves in $\mathbb{R}^2$ using the Brunn-Minkowski inequality. The argument may be found here: http://mathproblems123.wordpress.com/2012/05/09/minkowski-content-and-the-isoperimetric-inequality/

My question: How does one establish that equality holds if and only if the region is a disk? The "if" part is of course trivial, so really I am concerned about the "only if" part.

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    What's the equality case of the Brunn-Minkowski inequality? (Not a rhetorical question.)2012-05-30
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    I actually can't find a good discussion of when equality holds.2012-05-30
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    @QiaochuYuan: The equality holds if and only if the sets are homothetic.2012-10-30
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    More details can be found in the article: THE BRUNN-MINKOWSKI INEQUALITY of J. Gardner. Sorry I can't provide a good link for it.2012-10-30

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