Sections and subsets of simplexes

There is a locally convex space E and a compact simplex S ⊂ E with the following property: for any metrizable compact convex subset K of a locally convex space there is a subspace M ⊆ E such that K is affinely homeomorphic to M ∩ S. One possible choice is E = l₁ with the w topology induced by c and (Formula Presented) If X is a Banach space and S ⊂ X is a compact simplex, then for each s Ì 0 there is an operator T: X — X with finite dimensional range such that (Formula Presented) e for all x e S. Every infinite dimensional Banach space X contains a compact set K for which there is no bounded simplex S ⊂ X with K ⊂ S.