Let $U$ be an open, bounded, symmetric and convex subset of $\mathbb{R}^2$ such that $0 \in U$. Define $f: \mathbb{R}^2 \to [0, +\infty)$ as follows: $f(x,y) = \inf \{ \lambda \in \mathbb{R}_+ : (x,y) \in \lambda U \}$.
Prove that $f$ is a norm on $\mathbb{R}^2$.
It is easy to prove that $f(x,y) = 0 \Rightarrow (x,y) = (0,0)$ and that $f(\alpha x, \alpha y) = |\alpha|f(x,y)$. But how can I prove the triangle inequality?