I'm looking for a 1-homogeneous function $\pi \colon \mathbb{R}^n_{\geq 0} \to \mathbb{R}$ satisfying the following properties:
1) $\pi$ is not concave. This is equivalent to the fact that there exist $x, y \in\mathbb{R}^n_{\geq 0}$ such that $\pi(x+y) < \pi(x) + \pi(y)$.
2) For all $v = (v_1, \cdots , v_n) \in\mathbb{R}^n_{\geq 0}$ the quantity $\frac{i}{n}! \frac{\pi(v)^n}{\prod_{i}v_i}$ is bounded.
Any ideas?