# Metric Space Distances and Radiuses

Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realise X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?

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Metric 2

Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realise X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?

Firstly we will check if d is a metric.

1. for any

This is true, as 1 or 2.

2. , also true since if x and y are not equal, then 1 or 2.

3. - True

4.

This is true since maximum of is 2, while minimum of and .

In this case we get . True.

Now let T be a transformation of X into En.

Let be the distance between points x and y of En, and let Tx, Ty be any images of x and y, respectively.

If there is a length a >0 such ...

#### Solution Summary

Let X = {A, B, C, D} with d(A, D) = 2, but all the other distances equal to 1. Check that d is a metric. Prove that the metric space X is not isometric to any subset of En for any n. Can you realize X as a subset of a sphere S2 of appropriate radius, with the spherical 'great circle' metric?