RESEARCH ON GEOMETRY OF BANACH AND FUNCTION SPACES AND INEQUALITIES

研究了Banach空间和函数空间的几何结构，特别是与一些范数不等式的关系。研究了Banach空间的n-直和，这对于由凸函数n构造具有非l_p型范数的Banach空间是很重要的.主要结果如下。[1]On Banach空间和函数空间的几何结构及范数不等式：（1）推广了强随机克拉克森不等式，并刻划了所得不等式成立的Banach空间是p-一致光滑的空间，或等价地是强p型的空间，且所涉及的常数相同。(2)We引入了带权的Hanner型不等式，并用这些不等式刻画了2-一致光滑和2-一致凸空间。通过改变权的位置，得到的不等式刻画了p-一致光滑和q-一致凸空间。在L_p-空间中确定了这些不等式中权的最佳值。给出了这些Hanner型不等式的对偶定理。[2]We给出了Banach空间中的一个锐三角不等式及其逆不等式。[3]On（1）刻划了Banach空间的n-直和的以下性质：光滑性、弱近一致光滑性、WORTH性质、Schur性质等，作为应用，给出了几个非一致非方但具有不动点性质的Banach空间的例子。(2)We刻画了Banach空间的一致非l_n_1-凸性（B_n-凸性），特别是对Banach空间的l_1-和和l_∞-和. [4]在James常数、性质M与不动点性质的关系、Banach-Mazur距离与超自反性的关系等方面得到了一些结果。

Geometric structures of Banach and function spaces were investigated especially in relation with some norm inequalities. Also ψ-direct sums of Banach spaces were investigated, which seem important as we can easily construct many examples of Banach spaces with a non l_p-type norm from a convex function ψ. Major results are as follows.[1]On geometric structures of Banach and function spaces and norm inequalities :(1)We extended the strong random Clarkson inequality, and characterized the Banach spaces in which the resulting inequality is valid are those of p-uniform smooth, or equivalently those of strong type p with the same constants involved them.(2)We introduced Hanner-type inequalities with a weight and characterized 2-uniformly smooth, and 2-uniformly convex spaces with these inequalities. By changing the place of the weight, the resulting inequalities characterize p-uniformly smooth, and q-uniformly convex spaces. The best value of the weight in these inequalities were determined for L_p-spaces. A duality theorem for these Hanner-type inequalities was presented as well.[2]We presented a sharp triangle inequality and its reverse in a Banach space.[3]On ψ-direct sums of Banach spaces :(1)We characterized the following properties of ψ-direct sums of Banach spaces :smoothness, weak nearly uniform smoothness, WORTH property, Schur property etc. As an application we presented several examples of Banach spaces which are not uniformly non-square, but have the fixed point property.(2)We characterized uniform non l^n_1-ness (B_n-convexity) particularly for the l_1-sum and l_∞-sum of Banach spaces.[4]Some resuts were obatined on the James constant, on the relation between Property M and the fixed point property, and on the Banach-Mazur distance in relation with super-reflexivity etc.