For every convex body , there is a minimal matrix convex set , and a maximal matrix convex set , which have K as their ground level. We aim to find the optimal constant such that . For example, if is the unit ball in with the norm, then we find that This constant is sharp, and it is new for all . Moreover, for some sets K we find a minimal set L for which . In particular, we obtain that a convex body K satisfies 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 . We also introduce new explicit constructions of these (and other) dilations.