Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies

We extend Allard’s celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every $d$-dimensional varifold with locally bounded first variation and positive $d$-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane.