Some matrix rearrangement inequalities

We investigate a rearrangement inequality for pairs of n x n matrices: Let parallel to A parallel to(p) denote (Tr(A*A)(p/2))(1/p), the C-p trace norm of an n x n matrix A. Consider the quantity parallel to A+B parallel to(p)(p)+parallel to A-B parallel to(p)(p). Under certain positivity conditions, we show that this is nonincreasing for a natural "rearrangement" of the matrices A and B when 1 <= p <= 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 Harmer's inequality for LP 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 < infinity. At present this is known to be the case only for 1 < p <= 4/3, p = 2, and p >= 4. Several other rearrangement inequalities that are of interest in their own right are proved as the lemmas used in proving the main results.