Some matrix rearrangement inequalities

Abstract

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∥pp+∥A-B∥pp. 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