Shellable weakly compact subsets of $C[0,1]$

Abstract: We show that for every weakly compact subset $K$ of $C[0,1]$ with finite Cantor-Bendixson rank, there is a reflexive Banach lattice $E$ and an operator $T:E\rightarrow C[0,1]$ such that $K\subseteq T(B_E)$. On the other hand, we exhibit an example of a weakly compact set of $C[0,1]$ homeomorphic to $\omega^\omega+1$ for which such $T$ and $E$ cannot exist. This answers a question of M. Talagrand in the 80's.