Isoperimetric inequalities and calibrations

Abstract

The subject of these Notes is the new proof, proposed in [F. H{\'e}lein, In{\'e}galit{\'e} isop{\'e}rim{\'e}trique et calibrations, Annales de l'Institut Fourier 44, 4 (1994), 1211-1218] of the classical isoperimetric inequality in the plane. This proof is far from being the first one, but its interest is that it uses essentially integration by parts and Stoke's formula in a simple manner, like in a calibration. Here we expound again this proof and discuss in which sense our proof works like a calibration, or it can be understood in the framework of the theory of null Lagrangians, which goes back to Weierstrass, Mayer, Hilbert, Weyl, Carath{\'e}odory, and Lepage.