Rank distance
Three training points are (1,10), (3,200), (2.5,40) and the new point is (2,50). Compute the ranks, from 1 to n+1=4, in each coordinate and the rank distances. Which training point is the nearest neighbour? Show that the answer is unchanged if the second coordinate is replaced by its logarithm.
Coordinate 1, the values \{1,3,2.5\} plus the new value 2, sorts as 1<2<2.5<3. The ranks are 1\mapsto 1, new \mapsto 2, 2.5\mapsto 3, 3\mapsto 4. Coordinate 2, the values \{10,200,40\} plus the new value 50, sorts as 10<40<50<200. The ranks are 10\mapsto 1, 40\mapsto 2, new \mapsto 3, 200\mapsto 4. Rank pairs: (1,10) is (1,1); (3,200) is (4,4); (2.5,40) is (3,2); the new point is (2,3). The distances are
\begin{align*} \rho\bigl((1,10)\bigr)&=\max(|1-2|,|1-3|)=2,\\ \rho\bigl((3,200)\bigr)&=\max(|4-2|,|4-3|)=2,\\ \rho\bigl((2.5,40)\bigr)&=\max(|3-2|,|2-3|)=1. \end{align*}
The nearest neighbour is (2.5,40), with \rho=1. Taking logarithms of coordinate 2 does not change the ordering, hence not the ranks, hence not the distances.