On the Relaxation of a Class of Functionals Defined on Riemannian Distances

We study the relaxation of a class of functionals defined on distances induced by isotropic Riemannian metrics on an open subset of $\mathbb{R}^N$. We prove that isotropic Riemannian metrics are dense in Finsler ones and we show that the relaxed functionals admit a specific integral representation.