Let $p$, $q_1$ and $q_2$ be points in Euclidean space. Is the closest point to $p$ amongst $q_1$ and $q_2$ with the taxicab metric always the same as with the Euclidean metric ? In other words, do both metrics always produce the same ordering of distances ?
Motivation : I am looking to optimise a program that needs to compare distances very often, but where the actual value of said distances are unimportant as long as the order does not change.
With loss of generality , we can take $P = Origin$.
Let us make a Circle with $Center = P$ & Distance to $Q1$ is the radius.
Here , we can take the $angle = 45$ Degree.
We can also take $radius = 1$ , through scaling.
Euclidean Distance $ED(Q1)=1$ , while taxicab Distance $tc(Q1) = \sqrt{2}$.
Let us consider a line at $angle = 30$ Degree where $Q2$ may lie.
Let $Q2$ move from little inside the Circle to a Point on the Circle & then to little outside the Circle.
Here , $tc(Q2)$ varies on this line.
Intuitively :
We can tell which is nearer to $P$ , when looking at $ED(Q2)$.
We can NOT tell which is nearer to $P$ , when looking at $tc(Q2).
Numerically :
When $ED(PQ2) = 0.99$ , $tc(PQ2) = 0.99\sin{30}+0.99\cos{30} = 1.352365$
When $ED(PQ2) = 1.00$ , $tc(PQ2) = 1.00\sin{30}+1.00\cos{30} = 1.36602540$
When $ED(PQ2) = 1.01$ , $tc(PQ2) = 1.01\sin{30}+1.01\cos{30} = 1.3796856578$
Here , $tc(Q1) = 1.41421356$ , hence we can NOT tell which is nearer in Euclidean Distance , with the taxicab values.