@MikeMiller
let $U$ be an open convex neighborhood of $0$ in $V=\Bbb R^n$ such that $v \in U \Rightarrow -v \in U$ and such that $\partial U$ has the property that every ray from the origin interesects $\partial U$ exactly once.
Define $\|0\|=0$ and for any $v \in \Bbb R^n \setminus 0$, let $\|v\|=1/\alpha$ where $\alpha \in \Bbb R_{>0}$ is the unique constant such that $\alpha v \in \partial U$. Then clearly $\| \lambda v \|=\lambda \|v \|$ and since we have $-v \in U \Rightarrow v \in U$, we also have $v \in \partial U \Rightarrow -v \in \partial U$ which implies $\|-v\|=\|v\|$.