The convex point of continuity property in Banach spaces not containing ℓ1

We obtain a local characterization of the convex point of continuity property for every closed, bounded and convex subset not containing sequences equivalent to the standard unit basis of ℓ1. As a consequence, we prove, in the setting of Banach spaces without ℓ1-copies, that the convex point of continuity property is determined on subspaces with a Schauder basis, which is a partial answer to a well-known problem.