Michael Simon inequality is a fundamental tool in  geometric analysis and geometric measure theory.  Its extension to anisotropic integrands will allow to extend to anisotropic integrands a series of results which are currently known only for the area functional. 
In this talk I will present an anisotropic version of the Michael-Simon inequality, for for two-dimensional varifolds in R3, provided that the integrand is close to the area in the C1-topology. The proof is deeply inspired by posthumous notes by Almgren, devoted to the same result. Although our arguments overlap with Almgren’s, some parts are greatly simplified and rely on a nonlinear version of the planar multilinear Kakaeya inequality.