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
If X is a Banach space and S ⊂ X is a compact simplex, then for each 𝜖 > 0
there is an operator T : X → X with finite dimensional range such that
∥T(x) − x∥ < 𝜖 for all x ∈ S. Every infinite dimensional Banach space X
contains a compact set K for which there is no bounded simplex S ⊂ X with
K ⊂ S.
