  Containment problems for polytopes and spectrahedra appear in various
applications, such as linear and semidefinite programming, combinatorics,
convexity and stability analysis of differential equations. This paper explores
the theoretical background of a method proposed by Ben-Tal and Nemirovksi.
Their method provides a strengthening of the containment problem, that is
algorithmically well tractable. To analyze this method, we study abstract
operator systems, and investigate when they have a finite-dimensional concrete
realization. Our results give some profound insight into their approach. They
imply that when testing the inclusion of a fixed polyhedral cone in an
arbitrary spectrahedron, the strengthening is tight if and only if the
polyhedral cone is a simplex. This is true independent of the representation of
the polytope. We also deduce error bounds in the other cases, simplifying and
extending recent results by various authors.
