Finding the maximum/minimum of a homogeneous function on $\mathbb R^n$

Question

Suppose that $f:\mathbb R^n\to\mathbb R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?

Answer 1

I would say $x^*\in\mathop{\text{arg}\,\text{min}} f$, because it might be possible for other points $x$ to achieve $f(x)=0$. But otherwise, yes, the statement is true.

If $f(x^*)=0$, but $x\not\in\mathop{\text{arg}\,\text{min}} f$, then there must be some $x$ satisfying $f(x)<0$. But if that were the case, $f(\alpha x)\rightarrow-\infty$ as $\alpha\rightarrow\infty$. Thus $\inf_x f(x) = -\infty$, and $\mathop{\text{arg}\,\text{min}} f$ is empty—a contradiction.