Let $K \subset \mathbb{R}^2$ is closed convex set. Is $f(x) = \rho(x, K) = \min\limits_{k \in K}\rho(x,k)$ is also convex function?
I know that without the convexity assumption, this is not true. I can only prove reverse statement: if function $f$ is convex then $K$ is convex: $$0≤f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y).$$ And using that $\rho(x, y) = 0$ if $y\in K$. I can state that $f(λx+(1−λ)y) = 0$ and from this it follows that $λx+(1−λ)y \in A$.
But I have absolutely no idea to proof in the other direction. Can you help me with this question?
