Denote by DC the axiom of Dependent Choices. Given a normed space, we define the convex topology on E: its closed sets are the intersections of finite unions of closed convex sets. In ZFC (or even in ZF +HB where HB is the Hahn-Banach axiom), the convex topology is the weak topology on E. In ZF, the convex topology is intermediate between the weak topology and the strong one. Given a real number t>0, say that a sequence (a_n)_n of E is a t-sequence if for every no negative integer n, the distance between span(a_i : i < n) and the convex hull conv(a_i : i >=n) is > t. Say that a subset of E is J-reflexive if it does not contain any t-sequence for some t>0. R.C.James has proved in ZF +DC+HB that J-reflexive bounded closed convex subsets of a Banach space are weakly compact. We show in ZF+DC that J-reflexive bounded closed convex subsets of a Banach space are compact in the convex topology (whence they are weakly compact). We also show that the weak compactness of the closed unit ball of a (simply) reflexive Banach space does not imply BPI.