  Mean dimension for AH-algebras is introduced. It is shown that if a simple
unital AH-algebra with diagonal maps has mean dimension zero, then it has
strict comparison on positive elements. In particular, the strict order on
projections is determined by traces. Moreover, a lower bound of the mean
dimension is given in term of comparison radius. Using classification results,
if a simple unital AH-algebra with diagonal maps has mean dimension zero, it
must be an AH-algebra without dimension growth.
  Two classes of AH-algebras are shown to have mean dimension zero: the class
of AH-algebras with at most countably many extremal traces, and the class of
AH-algebras with numbers of extreme traces which induce same state on the
K0-group being uniformly bounded (in particular, this class includes
AH-algebras with real rank zero).
