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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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The equality holds in the Brunn Minkowski inequality if and only if the corresponding sets $A,B$ are homothetic. You can see that in the proof of the isoperimetric inequality used in the link, the Brunn Minkowski inequality is applied to $\Omega$ and a ball.

If we have equality in the isoperimetric inequality, then we will have equality in the Brunn Minkowski inequality which means that $\Omega$ is homothetic to a ball, hence $\Omega$ is a ball.

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    Not really. Strict Brunn-MInkowski inequality can be lost in the limit.2017-04-06
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    The argument is not really related to the one in the link... In $\Bbb{R}^2$ the best isoperimetric set must be convex. Therefore if $\Omega$ has the same area and the same perimeter as a disk $D$, then consider the Minkowski sum $\omega = 1/2\Omega+1/2D$. Then $\omega$ has the same perimeter as $D$ but its area is at least $|D|$ by Brunn Minkowski. But since $\Omega$ is optimal a quick argument shows that $|\omega|=|\Omega|=|D|$. Therefore we have equality in Brunn Minkowski and $\Omega$ is homothetic to $D$.2017-04-06
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    I'll have to think about it, but thanks! I didn't think about in this way.2017-04-07