Relative Isoperimetric Inequality in the Plane: the Anisotropic Case

We prove a relative isoperimetric inequality in the plane, when the perimeter is defined with respect to a convex, positively homogeneous function of degree one $H\colon\mathbb{R}^2 \rightarrow [0,+\infty[$. Under suitable assumptions on $\Omega$ and $H$, we also characterize the minimizers.