Different metrics yield different metric spaces

up:: Metric Space

Metric spaces with same set but different Metric Functions can be seen as different spaces altogether.

Consider, as in the image above, ${\mathbb{R}}^2$ with the euclidean metric

$$d_E(u,v)=\sqrt{\sum _{i=1}^n(u_i-v_i{)}^2}$$

and the taxi-cab metric

$$d_1(u,v)=\sum _i|u_i-v_i|$$

Note that $red$, $blue$ and $yellow$ distances are all equivalent shortest distances under the $L_1$ distance, while the shortest distance in the Euclidean metric is the $green$ path.

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References

• Metric space - Wikipedia