**Recall:** A function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is called $0$-homogeneous if
$f(\lambda x)= f(x)$ for every $\lambda>0$ and every $x\in \mathbb{R}^n$.

**Question:** Let $B$ a convex balanced and absorbent bounded domain of $\mathbb{R}^n$. Is the space of $0$-homogeneous $C^\infty(\mathbb{R}^n\setminus\{0\})$ functions dense in $H^1(\partial B)$?