  We investigate a rearrangement inequality for pairs of n-square matrices: Let
|A\|_p denote the C^p trace norm of an n-square matrix A. Consider the quantity
|A+B|_p^p + |A-B|_p^p. Under certain positivity conditions, we show that this
is nonincreasing for a natural ``rearrangement'' of the matrices A and B when 1
\le p \le 2. We conjecture that this is true in general, without any
restrictions on A and B. Were this the case, it would prove the analog of
Hanner's inequality for L^p function spaces, and would show that the unit ball
in C^p has the exact same moduli of smoothness and convexity as does the unit
ball in L^p for all 1 < p < \infty. At present this is known to be the case
only for 1 < p \le 4/3, p =2, and p\ge 4. Several other rearrangement
inequalities that are of interest in their own right are proved as the lemmas
used in proving the main results.
