Authors’ abstract: In this paper, we consider positively 1-homogeneous supremal functionals of the type \(F(u) := \sup_{\Omega} f(x,\nabla u(x))\). We prove that the relaxation \(\overline{F}\) is a difference quotient, that is, \[ \overline{F}(u) = R^{d_F}(u) := \sup_{x,y\in\Omega,\;x\neq y} \frac{u(x)-u(y)}{d_F(x,y)}\quad \text{for every }u \in W^{1,\infty}(\Omega), \] where \({d_F}\) is a geodesic distance associated to \(F\). Moreover, we prove that the closure of the class of 1-homogeneous supremal functionals with respect to \(\Gamma\)-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances.