距离函数的可微性研究

Studying on the Differentiability of Distance Functions

在一个赋范线性空间中,非空闭子集K的性质与距离函数d(x)的性质紧密相关。若X是一个Banach空间,K是X的非空闭凸集,X上的范数一致Gateaux(Frechet)可微,则d(x)在X的稠子集X\bdyK上是Gateaux(Frechet)可微。在一定的条件下,d(x)在X的每一点都是Gateaux(Frechet)可微。

In a real norm linear Space X,the properties of a non-empty closed set K are closely related to those of the distance functions d（x） which it generates.If K is convex,X has a uniformly Gateaux（uniformly Frechet） differentiable norm,then d is Gateaux（Frechet） differentiable on dense subset of X and even Gateaux differentiable on X.