A projective metric on a projective space is a metric on the underlying set such that shortest path with respect to this metric are parts of entire projective straight lines.
The 2-norm induces the Fubini-Study metric on a projective space (up to a real constant multiplier). Does any other p-norm also induce a projective metric on a projective space?
One idea is that a distance function like $d(x,y)=\operatorname{min}_{|\alpha|=1}||\frac{x}{||x||_q}-\alpha\frac{y}{||y||_q}||_p$ could work, either with $p=q$, or with $\frac{1}{p}+\frac{1}{q}=1$. But if one can show that neither distance function works, then that would also be valuable information.
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The linked Encyclopedia of Mathematics article about projective metric makes it sound like the problem of determining all projective metrics has been completely solved. So it seems to me that this question should have a definitive answer.