I have the following question: Let $ Y \subset \mathbb{R}^2 $ denote a finite set. Then, let us consider the squared distance function $ F:\mathbb{R}^2\rightarrow [0,\infty) $ defined by $ F(x)=\min_{y \in Y} ||x-y||_2^2 $.
Is $ F $ differentiable?
I mean, there are similar theorems, but all I found involve convexity of the set $ Y $, which is of course not given here. Normally, this convexity is then used to have a unique projection, an argument we cannot use for finite $ Y $.
So, how could a proof work in this setting?
Thank you very much!