Let $C$ be the set of functions $f:\mathbb{R}^n \to \mathbb{R}$ such that $x\mapsto f(x)$ and $x\mapsto \lVert x \rVert^2 - f(x)$ are both convex.
If $f$ belongs to $C$, then $f$ plus any affine function also does. What are the extreme points of $C$ modulo affine functions?
If $K$ is a convex domain of $\mathbb{R}^n$, the squared distance function to $K$ seems to be an extreme point, as well as $\lVert x \rVert^2$ minus that squared distance function. Are there any other extreme points?
