Abstract
We investigate a rearrangement inequality for pairs of n×n matrices: Let \(\|A\|_p\) denote (Tr(A*A)p/2)1/p, the Cp 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 Lp function spaces, and would show that the unit ball in Cp has the exact same moduli of smoothness and convexity as does the unit ball in Lp 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.
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