A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
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DOI：
10.1090/s0002-9947-1987-0902782-7
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发表时间：
1987-02
影响因子：
1.3
通讯作者：
J. Borwein;D. Preiss
中科院分区：
文献类型：
--
作者：
J. Borwein;D. Preiss
我们表明，通常，较低的Banach空间上的连续函数密集继承较低的次导数相同程度的光滑规范。特别地，具有Gâteaux（weak Hadamard，Frechet）光滑补的空间上的每个连续凸函数都是稠密Gâteaux（weak Hadamard，Frechet）可微的。我们的技术依赖于一个更强大的模拟Ekeland的变分原理，其中的功能是扰动的二次函数。这种“光滑”变分原理在非光滑分析问题中具有非常广泛的适用性。
We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Frechet) smooth renorm is densely Gâteaux (weak Hadamard, Frechet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.