Unique Properties of the Euclidean Norm

Question

Preamble
A friend and I enjoy passing the time by picking arbitrary favorites in mathematics. Recently we have been talking about our favorite distance metrics. Being an engineer by training, the Euclidean norm is close to my heart. Him, being a geometric group theorist, thinks that the euclidean norm is overrated. He favors the p-adic norm, or your common or garden taxicab metric.

Question
Does the euclidean norm possess any mathematical properties which are unique to it among metrics. Or are at least rare properties for a metric to possess. Features which make it in some way distinct as a norm.

Supplementary
If you have any personal favorite norms or recommendations for further reading I would appreciate hearing them.

There is a paper (link) about this in the context of economics. I don't find the authors' characterization of Euclidean distance very intuitive, but it's there. If you can come up with an intuitive explanation, let me know! — Casper, Apr 03 '21 at 10:02
Euclidean norm is indeed overrated, but why $p$-adic norm unless you are doing $p$-adic analysis on $\mathbb{Q}_p$ or similar? — user10354138, Apr 03 '21 at 10:38

score 1 · Answer 1 · answered Apr 03 '21 at 10:06

1

The generalization of the Euclidean norm to function spaces is the $L_2$ norm. Of all the $L_p$ spaces, only $L_2$ is a Hilbert space. [I leave it to the reader to track down the definitions of $L_p$ norm and of Hilbert space to see why this is a big deal.]

answered Apr 03 '21 at 10:06

Gerry Myerson

3,404

I agree. I recommend having a look at this recent discussion, where all elements of this "big deal" appear. The keyword is "orthogonality". – Giuseppe Negro Apr 03 '21 at 10:17

Unique Properties of the Euclidean Norm

1 Answers1