Proving a function $\rho:V\to[0,\infty)$ forms a norm on $V$ if and only if the unit ball $\{x \in V\mid ρ(x)

I have a function $\rho:V\to[0,\infty)$ where $V$ is a vector space and we know $\rho$ to be positive, absolutely homogeneous and non-degenerate (i.e.: we know $\rho$ satisfies all norm conditions other than the triangle inequality).

I need to prove that $\rho$ forms a norm on $V$ (i.e.: satisfies the triangle inequality) if and only if the unit ball $\{x\in V\mid \rho(x)<1\}$ is convex.

I've shown that if the unit ball is convex, we have $tx+(1−t)y<1$ for $x,y\in V,t\in[0,1],$ but I am unsure as to where to go from there, or if that's even useful.

I've seen a similar question like this on here, but it is for $V=\mathbb{R}^2$, with the proof taking $x\in V$ to be a scalar, and I don't know if I can do that for a general vector space.

Any help is appreciated.