Isoperimetric Inequalities and Calibrations

  • Fr Ed, Eric H Elein
  • Published 2007

Abstract

The subject of these Notes is a new proof, proposed in 4], of the classical isoperimetric inequality in the plane. This proof is far from being the rst one, and we may refer to 1] for a small review of some various proofs which are known, and for further references. To my opinion, the interest of this proof is that it uses essentially integration by parts and Stokes' formula in a simple manner, like in a calibration. Let us explain it brieey: consider a smooth domain of the plane R 2 , and let x, y denote points of @, the boundary of. We denote by t y the unit tangential vector to @ at y, such that, if n y is the exterior normal vector to @ at y, then (n y ; t y) forms a direct basis. We then let for x 6 = y We rst x y and t y and we build the 1-form (see Figure 1) = hV (y; t y ; x); dxi: We integrate over x 2 @. Using the fact that V is of norm 1 everywhere, and Stokes' formula, we get j@j Z @ = Z x2 dd = Z x2 2 det(y ? x; t y) jx ? yj 2 dx 1 ^ dx 2 :

Cite this paper

@inproceedings{Ed2007IsoperimetricIA, title={Isoperimetric Inequalities and Calibrations}, author={Fr Ed and Eric H Elein}, year={2007} }