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Which of the following define a metric?

a. $d((x, y), (x’, y’)) = \min\{|x – x’|, |y – y’|\}$ on $\mathbb{R}^2$.
b. $d((x, y), (x’, y’)) = |x| + |y| + |x’| + |y’|$ on $\mathbb{R}^2$..
c. $D((x, y), (x’, y’)) = d(x, x’) + d(y, y’)$ on $X \times X$, where $(X, d)$ is a metric space.

I have found that (a) is not true, but not sure about the others.

copper.hat
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user55554
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  • Usually the easiest thing to check when considering possible metrics is that $d(x, x) = 0$. The next easiest is that $d(x, y) = d(y, x)$ and maybe non-negativity. It is often trickier to determine whether $d(x, y) = 0 \implies x=y$ and the triangle inequality are true. What have you determined about the proposed metric in b? – Michael Albanese Jan 09 '13 at 07:13
  • Try $(x,y)=(1,1),(x',y')=(1,1)$ for b). Try proving c) from the axioms. – copper.hat Jan 09 '13 at 07:13
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    -1: As pointed out in the answer below the only thing you have to do is to go through a list of axioms and try to verify them. Surely a little bit of thinking results in more than "I have found that (a) is not true, but not sure about the others." Also see: http://meta.math.stackexchange.com/q/1803/ – Martin Jan 09 '13 at 07:18

2 Answers2

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To show that a distance formula is a metric, you need to show that

1) $d(x, y) \geq 0$ (non-negativity)
2) $d(x, y) = 0 \iff x=y$
3) $d(x, y) = d(y, x)$ (symmetry)
4) $d(x, z) \leq d(x, y) + d(y, z)$ (triangle inequality)

Properties 1, 2, 3 are often easily established or checked.

Conversely, to show that a distance formula is not a metric, you need to show that one of these properties does not hold. Likewise, it is often property 4 that tricky to violate.

On to your question:

a) This is not a metric, which you stated. Which property does it violate?

b) This is not a metric. Which property does it violate?

c) This is known as the taxicab metric, or Manhattan metric, when $X =\mathbb{R}$. Prove that it is a metric.

Calvin Lin
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  • Also known as the Manhattan length. – copper.hat Jan 09 '13 at 07:20
  • @copper.hat: I face the taxicab metric while I teach Geometry. Why is it really called like that? Taxi?? Thanks. – Mikasa Jan 09 '13 at 07:45
  • @BabakSorouh: http://en.wikipedia.org/wiki/Taxicab_geometry – Martin Jan 09 '13 at 07:55
  • @Babak Because that is how taxis in Manhattan calculate distance. The streets of Manhattan are laid out in a (almost) perfect grid, and so they will say "I'm X blocks away", where X is the sum of the horizontal and vertical distance. It doesn't really matter what path they take to get there, the distance travelled is (almost) the same. – Calvin Lin Jan 09 '13 at 07:55
  • @Martin: Thanks. Thanks Calvin. Nice illustration. – Mikasa Jan 09 '13 at 07:57
  • Wiring on integrated circuits used to be laid out in orthogonal line segments, hence the $l_1$ metric (Manhattan) was a good estimate of wire length (which affected signal delay). – copper.hat Jan 09 '13 at 08:12
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For (b) try seeing what happens when $(x,y)=(x',y')$.

emka
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