Some matrix rearrangement inequalities

We investigate a rearrangement inequality for pairs of n×n matrices: Let ∥A∥_p denote (Tr(A* A) ^p/2)^1/p , the C ^p trace norm of an n×n 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≤ 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 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<∞. 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.