Which of the following define a metric?

Question

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.

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

Answer 1

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.

Answer 2

For (b) try seeing what happens when $(x,y)=(x',y')$.