Matrix convexity, Choquet boundaries and Tsirelson problems

Following work of Evert, Helton, Klep and McCullough on free LMI domains, we ask when a matrix
convex set is the closed convex hull of its Choquet points. This is a finite-dimensional version
of Arveson's non-commutative Krein-Milman theorem, and in fact some matrix convex sets can fail
to have any Choquet points. The general problem of determining whether a given matrix convex set
has this property turns out to be difficult because for certain matrix convex sets this is
equivalent to a weak version of Tsirelson's problem. This weak variant of Tsirelson's problem is
known to be equivalent to Connes' embedding conjecture, and is considered a hard problem by many
experts. Our approach provides new geometric variants of Tsirelson type problems for pairs of
convex polytopes which may be easier to rule out than Tsirelson's original problems.