Minkowski functional of balanced domain with smooth boundary

Question

Let $D \subset \mathbb{C}^n$ be a bounded balanced domain whose boundary is $\mathcal{C}^2$-smooth. Here, balanced means that if $z \in D$ and $|\lambda| \leq 1$ implies that $\lambda z \in D$. We define the Minkowski functional of $D$ to be the function $h: \mathbb{C}^n \to \mathbb{R}_+$ given by $z \mapsto \inf \{ t \in \mathbb{R}_+ : z/t \in D\}$. Is it true that $h$ is $\mathcal{C}^2$-smooth on $\mathbb{C} \setminus \{0\}$?