Reference request: Nonsmoothness of the projection operator

Question

Let $C\subset\mathbb{R}^N$ be nonempty closed and convex, and let $P_C\colon\mathbb{R}^N\to\mathbb{R}^N$ be its projection operator. I've heard it is well-known that $P_C$ in general is not a smooth operator (i.e. if you think of $P_C(x) = (f_1(x), f_2(x), ..., f_N(x))$, where each $f_i\colon\mathbb{R}^N\to\mathbb{R}$, then the Jacobian will not be defined everywhere since some of those $(f_i)_{1\leq i\leq N}$ will be nonsmooth). This can be seen even for simple examples in $\mathbb{R}$ (e.g. $C=[0,1]$). I am looking for a journal article which points out this simple fact -- that such operators can be nonsmooth.

Answer 1

A. Shapiro. Directionally nondifferentiable metric projection. J. Optim. Theory Appl., 81(1):203–204, 1994. https://doi.org/10.1007/BF02190320

constructs a convex set in $\mathbb R^2$, such that $P_C$ is not even directionally differentiable at one point.