 Convex-block separation criterion for reflexivity

A Banach space $X$ is reflexive exactly when, for every $\theta >0$ and every sequence $(x_i)$ in $B_X$, some convex combination of a finite initial segment lies within $\theta$ of a convex combination of the remaining tail. In a reflexive space both convex hulls approximate a common weak cluster point. The converse is the convex-block form of the James reflexivity criterion.