  To every convex body $K \subseteq \mathbb{R}^d$, one may associate a minimal
matrix convex set $\mathcal{W}^{\textrm{min}}(K)$, and a maximal matrix convex
set $\mathcal{W}^{\textrm{max}}(K)$, which have $K$ as their ground level. The
main question treated in this paper is: under what conditions on a given pair
of convex bodies $K,L \subseteq \mathbb{R}^d$ does
$\mathcal{W}^{\textrm{max}}(K) \subseteq \mathcal{W}^{\textrm{min}}(L)$ hold?
For a convex body $K$, we aim to find the optimal constant $\theta(K)$ such
that $\mathcal{W}^{\textrm{max}}(K) \subseteq \theta(K) \cdot
\mathcal{W}^{\textrm{min}}(K)$; we achieve this goal for all the $\ell^p$ unit
balls, as well as for other sets. For example, if $\overline{\mathbb{B}}_{p,d}$
is the closed unit ball in $\mathbb{R}^d$ with the $\ell^p$ norm, then \[
\theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p - 1/2|}. \] This constant is
sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a
minimal set $L$ for which $\mathcal{W}^{\textrm{max}}(K) \subseteq
\mathcal{W}^{\textrm{min}}(L)$. In particular, we obtain that a convex body $K$
satisfies $\mathcal{W}^{\textrm{max}}(K) = \mathcal{W}^{\textrm{min}}(K)$ if
and only if $K$ is a simplex.
  These problems relate to dilation theory, convex geometry, operator systems,
and completely positive maps. We discuss and exploit these connections as well.
For example, our results show that every $d$-tuple of self-adjoint operators of
norm less than or equal to $1$, can be dilated to a commuting family of
self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit
constructions of these (and other) dilations.
