lp distance, p tends to minus infinity

Question

Minkowki distance as p tends to minus infinity equals to the smallest difference along any coordinate dimension. Is this difference a metrics? Even if the lp norm with p tending to minus infinity is not a norm?

Answer 1

No, the function $\rho(x,y) = \min_i(|x_i-y_i|)$ is not a metric. For one thing, positivity fails: $\rho $ turns into zero whenever $x_i=y_i$ for some $i$, even though $x$ and $y$ may still be distinct. More seriously, the triangle inequality fails: e.g., $$ \rho((1,1), (9,9)) > \rho((1,1), (1,8))+\rho((1,8), (9,9)) $$ ($8>1+1$)