Is a positively homogeneous convex function lower semi-continuous?

Question

Here the definition of positively homogeneous function $f$ is \begin{equation} f(\lambda x) = \lambda f(x), \quad \lambda \ge 0, \end{equation} and $f$ is defined on linear space $\mathbb{R}^n$.

I think this statement could not be held, but I could not find any counter example.

Answer 1

Define $f:\mathbb{R}^2 \to [0,\infty]$ by $f(x) = \begin{cases} 0,& x \in (-\infty,0)^2 \cup \{(0,0)\}\\ \infty,& \text{otherwise} \end{cases}$.

Then let $x_n = (-{1 \over n},-1)$, we have $f(x_n) = 0$, $\liminf_n f(x_n) < f(\lim_n x_n) = \infty$.

Hence $f$ is convex but not lsc.