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.