### 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∥_{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.

Original language | English (US) |
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Pages (from-to) | S315-S324 |

Journal | Annali di Matematica Pura ed Applicata |

Volume | 185 |

Issue number | SUPPL. 5 |

DOIs | |

State | Published - Jan 1 2006 |

### All Science Journal Classification (ASJC) codes

- Applied Mathematics

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## Cite this

*Annali di Matematica Pura ed Applicata*,

*185*(SUPPL. 5), S315-S324. https://doi.org/10.1007/s10231-004-0147-z