Strongly Convex Functions, Moreau Envelopes, and the Generic Nature of Convex Functions with Strong Minimizers
该文在有限维空间中为下半连续凸函数定义了一个完备度量，使得函数序列的收敛等价于上图收敛，并证明了强凸函数集是稠密但第一纲的，而具有强极小点的凸函数集是第二纲的。
In this work, using Moreau envelopes, we define a complete metric for the set of proper lower semicontinuous convex functions in a finite-dimensional space. Under this metric, the convergence of each sequence of convex functions is epi-convergence. We show that the set of strongly convex functions is dense but it is only of the first category. On the other hand, it is shown that the set of convex functions with strong minima is of the second category.