Simultaneous approximation from convex sets

We characterize best simultaneous approximation from convex sets, in terms of onesided Gateaux derivatives, using the ℓ1 and ℓ∞ vectorial norms, in a general setting and in L1. In L1, we give necessary and sufficient conditions for uniqueness. We show that p-type modulus of convexity implies order p-strong unicity of the best ℓ∞ simultaneous approximation.