The Question: Let $X=\mathbb{C}^n$ and $0<p<1$. Define $d_p$ by $d_p(z,w)= (\sum \limits_{k=1}^{n} |z_k-w_k|^p)^\frac{1}{p}$, where $z=(z_1,z_2,\ldots, z_n)$ and $w=(w_1, w_2, \ldots, w_n)$ are in $\mathbb{C}^n$. Does $d_p$ define a metric on $\mathbb{C}^n$. My Work: I have been playing around with factoring these sums out and I think I have came up with why it fails. I have found that for $0<p<1$ you actually end up with the reciprocal of Minkowski's Inequality, whcih would fail the triangle inequality. Am I on the right track? Thanks for any hints and help!
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If $0<p<1$, $d_{p}(\cdot)$ does not define a metric; rather, it defines what is known as a "quasi-metric" in the sense that $d(x,z)\leq C(d(x,y)+d(y,z))$ for some constant $C\geq1$. This has to do with the fact that the function $|x|^{p}$ is concave for $p<1$, whereas when $p\geq1$ it is convex (and so one can take $C=1$ in that case). If you discard the constant $C$, you end up with a reverse triangle inequality: $d(x,z)\geq d(x,y)+d(y,z)$. |
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