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From the answer to this question, we have that there exist absolutely homogenous, non translation invariant metrics on $\mathbb{R}^2$ that induce a norm. Does there exist a metric space $(\mathbb{R}^n, d)$ where $d$ is absolutely homogenous but that doesn’t induce a norm $p:\mathbb{R}^n\to \mathbb{R}$ by $p(x)=d(0,x)$. If not, why not?

Luke
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    The exact same thing works in any dimension – Didier Nov 30 '23 at 19:50
  • I’m asking for one that doesn’t induce a norm. – Luke Nov 30 '23 at 19:52
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    I was referring to the answer in the question you linked. It presents a metric that is homogeneous and does not come from a norm – Didier Nov 30 '23 at 20:15
  • Right, but that’s not what I’m asking about. The metric presented does still define a norm by considering d(x,0), even though it isn’t the metric induced by that norm. I’m asking if there’s a similar metric that doesn’t define a norm by considering distance from 0. – Luke Nov 30 '23 at 23:17

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