Steiner symmetrization preserves area?
Solution 1:
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