  Melchior's inequality implies that the average line-length in a simple,
rank-$3$, real-representable matroid is less than $3$. A similar result holds
for complex-representable matroids, using Hirzebruch's inequality, but with a
weaker bound of $4$. We show that the average plane-size in a simple, rank-$4$,
complex-representable matroid is bounded above by an absolute constant, unless
the matroid is the direct-sum of two lines. We also prove that, for any integer
$k$, in complex-representable matroids with rank at least $2k-1$, the average
size of a rank-$k$ flat is bounded above by a constant depending only on $k$.
Finally, we prove that, for any integer $r\ge 2$, the average flat-size in
rank-$r$ complex-representable matroids is bounded above by a constant
depending only on $r$. We obtain our results using a theorem, due to Ben Lund,
that gives a good estimate on the number of rank-$k$ flats in a
complex-representable matroid.
