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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:

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.) – Qiaochu Yuan May 30 '12 at 22:33
I actually can't find a good discussion of when equality holds. – Potato May 30 '12 at 22:39
@QiaochuYuan: The equality holds if and only if the sets are homothetic. – Beni Bogosel Oct 30 '12 at 20:52
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. – Beni Bogosel Oct 30 '12 at 20:55

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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