If all bounded linear
operators from L1 into a Banach space X are Dunford-Pettis (i.e. carry weakly
convergent sequences onto norm convergent sequences), then we say that X has
the complete continuity property (CCP). The CCP is a weakening of the
Radon-Nikodým property (RNP). Basic results of Bourgain and Talagrand began to
suggest the possibility that the CCP, like the RNP, can be realized as an internal
geometric property of Banach spaces; the purpose of this paper is to provide such a
realization. We begin by showing that X has the CCP if and only if every bounded
subset of X is Bocce dentable, or equivalently, every bounded subset of X
is weak-norm-one dentable (§2). This internal geometric description leads
to another; namely, X has the CCP if and only if no bounded separated
δ-trees grow in X, or equivalently, no bounded δ-Rademacher trees grow in X
(§3).
