Would this be a metric?

Question

I just read about the taxicab metric defined on $\mathbb{R}^2$.

Suppose you have the plane with fixed $x$-axes and $y$-axes. A path from point A and B on the plane must satisfy: you can only move in certain directions, say the $0$, $120$, or $240$ degree directions (wrt positive $x$-axis) and take finite number of turns.

If you defined $d(A,B)$ to be the infimum for such paths from $A$ to $B$. Do you get a metric?

Perhaps you would be interested in learning about "normed spaces". Taxicab metric and yours are two good examples. — GEdgar, Apr 16 '13 at 14:00

Cameron Buie · Answer 1 · 2013-04-16T14:55:37.820

1

No. It will not be symmetric as stated. For example, if $A=\langle 0,0\rangle$ and $B=\langle -1,0\rangle$, then $d(A,B)=2,$ but $d(B,A)=1$.

edited Apr 16 '13 at 14:55

answered Apr 16 '13 at 14:38

Cameron Buie

102,994

How do you get d(B,A) = 1 ? In 1 move, to go from B to A is impossible in these conditions because it is in the direction 90 degrees. – Gautam Shenoy Apr 16 '13 at 14:53
Oops! I transposed the coordinates. Fixed. – Cameron Buie Apr 16 '13 at 14:56
OK, I understood 0, 120, or 240 in either direction. Your interpretation may be more likely, but either direction is needed for a metric. – GEdgar Apr 16 '13 at 15:10

score 0 · Answer 2 · answered Apr 16 '13 at 13:47

0

Yes, because:

1) It would be definite positive.

2) It will be symmetric, because you can retrace your path back.

3) It will satisfy the triangle inequality because it will inherit this property from the norm defined in $\mathbb{R}^2$.

answered Apr 16 '13 at 13:47

Ambesh

3,312

1

I think the triangle inequality needs more reasoning. – GEdgar Apr 16 '13 at 14:01
@GEdgar: True, but the symmetry is the big issue. Yes, Mike, you can retrace your path back, but that's not to say that it will be the best path, so symmetry need not hold. See my answer. – Cameron Buie Apr 16 '13 at 14:41
Because you can trace ur path merely implies that the length of path traced is an upper bound to the reverse path. Not necessarily equal. – Gautam Shenoy Apr 16 '13 at 15:00
I think $d(A,B) \leq 3$ – Gautam Shenoy Apr 16 '13 at 15:05

Would this be a metric?

2 Answers2