  We report on measurements of the local friction law at a multi-contact
interface formed between a smooth rubber and statistically rough glass lenses,
under steady state friction. Using contact imaging, surface displacements are
measured, and inverted to extract both distributions of frictional shear stress
and contact pressure with a spatial resolution of about 10~$\mu$m. For a glass
surface whose topography is self-affine with a Gaussian height asperity
distribution, the local frictional shear stress is found to vary strongly
sub-linearly with the local contact pressure over the whole investigated
pressure range. Such sub-linear behavior is also evidenced for a surface with a
non Gaussian height asperity distribution, demonstrating that, for such
multi-contact interfaces, Amontons-Coulomb's friction law does not prevail at
the local scale.
