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In this paper we consider positively 1-homogeneous supremal functionals of the type $$ F(u) := {\rm sup}_{\Omega}f(x,\nabla u(x))$$. We prove that the relaxation $\bar{F}$ is a difference quotient, that is $$ \bar{F}(u) = R^{d_F}(u): = \mathop{\rm sup}_{x,y\in\Omega,x\neq y}\frac{u(x)-u(y)}{d_F(x,y)}\quad \mbox{for every}\ u\in W^{1,\infty}(\Omega),$$ 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 Γ-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.