A counterexample to a problem on points of continuity in Banach spaces

In a previous paper of the first two authors [GM] the space $J{T_\infty }$ was constructed as a James space over a tree with infinitely many branching points. It was proved that the predual ${B_\infty }$ of $J{T_\infty }$ fails the "point of continuity property."

In the present paper we show that ${B_\infty }$ has the so-called "convex point of continuity property" thus answering a question of Edgar and Wheeler [EW] in the negative.