  This article resides in the realm of the noncommutative (free) analog of real
algebraic geometry - the study of polynomial inequalities and equations over
the real numbers - with a focus on matrix convex sets $C$ and their projections
$\hat C$. A free semialgebraic set which is convex as well as bounded and open
can be represented as the solution set of a Linear Matrix Inequality (LMI), a
result which suggests that convex free semialgebraic sets are rare. Further,
Tarski's transfer principle fails in the free setting: The projection of a free
convex semialgebraic set need not be free semialgebraic. Both of these results,
and the importance of convex approximations in the optimization community,
provide impetus and motivation for the study of the free (matrix) convex hull
of free semialgebraic sets.
  This article presents the construction of a sequence $C^{(d)}$ of LMI domains
in increasingly many variables whose projections $\hat C^{(d)}$ are
successively finer outer approximations of the matrix convex hull of a free
semialgebraic set $D_p=\{X: p(X)\succeq0\}$. It is based on free analogs of
moments and Hankel matrices. Such an approximation scheme is possibly the best
that can be done in general. Indeed, natural noncommutative transcriptions of
formulas for certain well known classical (commutative) convex hulls does not
produce the convex hulls in the free case. This failure is illustrated on one
of the simplest free nonconvex $D_p$.
  A basic question is which free sets $\hat S$ are the projection of a free
semialgebraic set $S$? Techniques and results of this paper bear upon this
question which is open even for convex sets.
