  This article shows the existence of a class of closed bounded matrix convex
sets which do not have absolute extreme points. The sets we consider are
noncommutative sets, $K_X$, formed by taking matrix convex combinations of a
single tuple $X$. In the case that $X$ is a tuple of compact operators with no
nontrivial finite dimensional reducing subspaces, $K_X$ is a closed bounded
matrix convex set with no absolute extreme points.
  A central goal in the theory of matrix convexity is to find a natural notion
of an extreme point in the dimension free setting which is minimal with respect
to spanning. Matrix extreme points are the strongest type of extreme point
known to span matrix convex sets; however, they are not necessarily the
smallest set which does so. Absolute extreme points, a more restricted type of
extreme points that are closely related to Arveson's boundary, enjoy a strong
notion of minimality should they span. This result shows that matrix convex
sets may fail to be spanned by their absolute extreme points.
