Research on geometric structures of Banach and function spaces with applications

我们得到了关于Banach空间几何结构的一系列结果，特别是关于它们的ψ直接和。主要结果如下。我们展示了von Neumann-Jordan常数和James常数之间的一个非常简单的不等式，它改进了之前关于它们关系的所有结果。我们还展示了另一个不等式，它对前面的结果给出了统一的看法。对于包括l_∞和在内的ψ-直接和，我们刻画了一致的非l^n1性和弱的近似一致光滑性（对于有限多个Banach空间）。作为应用，给出了关于不动点性质和超自反性的一些结果。尖锐平均三角不等式也得到了证明。

We obtained a sequence of results on geometric structures of Banach spaces, especially for their ψ-direct sums. Major results are as follows. We showed a quite simple inequality between the von Neumann-Jordan and James constants, which refined all the previous results on their relation. We also showed another inequality which gives a unifying sight on these previous results. For ψ-direct sums, including the l_∞-sum, we characterized the uniform non-l^n1-ness, and weak nearly uniform smoothness (for finitely many Banach spaces). As applications some results on the fixed point property and super-reflexivity were given. Sharp mean triangle inequality was shown as well.