Minimal and maximal matrix convex sets

For every convex body K⊆R^d, there is a minimal matrix convex set W^min(K), and a maximal matrix convex set W^max(K), which have K as their ground level. We aim to find the optimal constant θ(K) such that W^max(K)⊆θ(K)⋅W^min(K). For example, if B‾_p,d is the unit ball in R^d with the ℓ^p norm, then we find that θ(B‾_p,d)=d^{1−|1/p−1/2|}. This constant is sharp, and it is new for all p≠2. Moreover, for some sets K we find a minimal set L for which W^max(K)⊆W^min(L). In particular, we obtain that a convex body K satisfies W^max(K)=W^min(K) only if K is a simplex. These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every d-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most d. We also introduce new explicit constructions of these (and other) dilations.