几个矩阵重排不等式
Some Matrix Rearrangement Inequalities
摘要：
本文研究了n-方阵对的一个重排不等式：设|A p表示n-方阵A的C^p迹范数.考虑数量|A+B|_p^p +| A-B|在一定的正性条件下，我们证明了当1 \le p \le 2.我们猜想，这在一般情况下是正确的，对A和B没有任何限制。如果是这样的话，它将证明L^p函数空间中类似的汉纳不等式，并将证明对于所有1 < p < \infty，C^p中的单位球与L^p中的单位球具有完全相同的光滑性和凸性模。目前，已知仅对于1 < p \le 4/3，p =2和p \ge 4是这种情况。其他几个重排不等式，是在自己的权利感兴趣的证明作为引理用于证明的主要结果。
英文摘要：
We investigate a rearrangement inequality for pairs of n-square matrices: Let |A\|_p denote the C^p trace norm of an n-square 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 \le p \le 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 < \infty. At present this is known to be the case only for 1 < p \le 4/3, p =2, and p\ge 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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