  A matrix convex set is a set of the form $\mathcal{S} = \cup_{n\geq
1}\mathcal{S}_n$ (where each $\mathcal{S}_n$ is a set of $d$-tuples of $n
\times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and
under formation of direct sums. We study the geometry of matrix convex sets and
their relationship to completely positive maps and dilation theory. Key
ingredients in our approach are polar duality in the sense of Effros and
Winkler, matrix ranges in the sense of Arveson, and concrete constructions of
scaled commuting normal dilation for tuples of self-adjoint operators, in the
sense of Helton, Klep, McCullough and Schweighofer.
  Given two matrix convex sets $\mathcal{S} = \cup_{n \geq 1} \mathcal{S}_n,$
and $\mathcal{T} = \cup_{n \geq 1} \mathcal{T}_n$, we find geometric conditions
on $\mathcal{S}$ or on $\mathcal{T}$, such that $\mathcal{S}_1 \subseteq
\mathcal{T}_1$ implies that $\mathcal{S} \subseteq C\mathcal{S}$ for some
constant $C$.
  For instance, under various symmetry conditions on $\mathcal{S}$, we can show
that $C$ above can be chosen to equal $d$, the number of variables, and in some
cases this is sharp. We also find an essentially unique self-dual matrix convex
set $\mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion
and dilation results hold with constant $C=\sqrt{d}$.
  Our results have immediate implications to spectrahedral inclusion problems
studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do
not depend on the ranks of the pencils determining the free spectrahedra in
question, but rather on the "number of variables" $d$. There are also
implications to the problem of existence of (unital) completely positive maps
with prescribed values on a set of operators.
