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I just finished reading (and understanding) Steiner's proof of the isoperimetric inequality. His proof (which is sadly incomplete) seems to rely much on the fact Steiner symmetrization preserves area and decreases perimeter.

The only resource I've found which provides proof of those two things are Andrejs Treibergs' beamer slides , however ,they are confusing as he is a bit heavy on notation.

Can someone please point me in the right direction on where I could find an alternative resource providing detailed proofs. If possible, a proof if you don't know a resource would be highly appreciated.

Adeeb
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    This is just by the construction of Steiner symmetric action and Fubini's theorem( where you can get: in order to get area, you integrate length of chords with respect to proper measure) – henryforever14 Apr 08 '13 at 03:00

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As henryforever14 noted, the preservation of area is not hard to prove: simply apply Fubini's theorem to the characteristic function of the set. The fact that perimeter does not increase is not nearly as easy. I recommend the sources

  • Geometric inequalities by Burago and Zalgaller
  • Isoperimetric inequalities in mathematical physics by Pólya and Szegő
  • The perimeter inequality under Steiner symmetrization: cases of equality (Ann. of Math 162 (2005), 525-555) by M. Chlebík, A. Cianchi, and N. Fusco
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