# Prove that d is a METRIC

kingwinner

## Homework Statement

Let (X,ρ) and (Y,σ) be metric spaces.
Define a metric d on X x Y by d((x1,y1),(x2,y2))=max(ρ(x1,x2),σ(y1,y2)).
Verify that d is a metric.

## The Attempt at a Solution

I proved positive definiteness and symmetry, but I am not sure how to prove the "triangle inequality" property of a metric. How many cases do we need in total, and how can we prove it?

Any help is appreciated!

kingwinner
So to verify the triangle inequality, we need to prove that
max(ρ(x1,x2),σ(y1,y2))≤ max(ρ(x1,x3),σ(y1,y3)) + max(ρ(x3,x2),σ(y3,y2)) for ANY (x1,y1),(x2,y2),(x3,y3) in X x Y.

How many separate cases do we need? I have trouble counting them without missing any...Is there a systematic way to count?

Case 1: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=ρ(x1,x3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

This case is simple, the above inequality is true since ρ is a metric.

Case 2: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=σ(y1,y3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

For example, how can we prove case 2?

Any help is appreciated!

Last edited:
Tinyboss
Suppose that $$\rho(x_1,x_2)\ge\sigma(y_1,y_2)$$. What do you know about $$\rho(x_1,x_3)+\rho(x_3,x_2)$$? Can you infer anything about the right-hand side of your inequality based on that?

kingwinner
Case 2: max(ρ(x1,x2),σ(y1,y2))=ρ(x1,x2), max(ρ(x1,x3),σ(y1,y3))=σ(y1,y3), max(ρ(x3,x2),σ(y3,y2)) =ρ(x3,x2)

Suppose that $$\rho(x_1,x_2)\ge\sigma(y_1,y_2)$$. What do you know about $$\rho(x_1,x_3)+\rho(x_3,x_2)$$? Can you infer anything about the right-hand side of your inequality based on that?
We'll have ρ(x1,x3)+ρ(x3,x2) ≥ σ(y1,y2)).

But I think for case 2, we need to prove that ρ(x1,x2)≤σ(y1,y3)+ρ(x3,x2) instead?? How can we prove it?

Thanks!

kingwinner
$$\max(\rho(x_1,x_2),\sigma(y_1,y_2))\le\max(\rho(x_1,x_3),\sigma(y_1,y_3))+\max(\rho(x_3,x_2),\sigma(y_3,y_2))$$.
Suppose $$\rho(x_1,x_2)\ge\sigma(y_1,y_2)$$. Since $$\rho$$ is a metric, you know that $$\rho(x_1,x_3)+\rho(x_3,x_2)\ge\rho(x_1,x_2)$$.
So what do you know about $$\max(\rho(x_1,x_3),\square)+\max(\rho(x_3,x_2),\square)$$, regardless of what's in the squares? You know it's at least as big as $$\rho(x_1,x_2)$$.