  We study containment and uniqueness problems concerning matrix convex sets.
First, to what extent is a matrix convex set determined by its first level? Our
results in this direction quantify the disparity between two product
operations, namely the product of the smallest matrix convex sets over $K_i
\subseteq \mathbb{C}^d$, and the smallest matrix convex set over the product of
$K_i$. Second, if a matrix convex set is given as the matrix range of an
operator tuple $T$, when is $T$ determined uniquely? We provide counterexamples
to results in the literature, showing that a compact tuple meeting a minimality
condition need not be determined uniquely, even if its matrix range is a
particularly friendly set. Finally, our results may be used to improve dilation
scales, such as the norm bound on the dilation of (non self-adjoint)
contractions to commuting normal operators, both concretely and abstractly.
