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We all know and love the standard $1,2,$ and $\infty$-norms on $\mathbb{R}^n$. However, I have never seen anyone mention uses for any of the other $k$-norms that I'm defining as

$$|x|_k=\left(\sum_{i=1}^n|x_i|^k\right)^{1/k}$$

Where $x$ is some vector in $\mathbb{R}^n$. Are there any practical uses for other norms? I know that all the norms are equivalent in some sense, and why we do use the ones I mentioned (as in this question: Why do we use the Euclidean metric on $\mathbb{R}^2$?), but my question is whether there are any uses for, say, the $3$-norm or any others?

I notice that there are a lot of other questions that dance around this one, but never ask it, so if I missed one and this is a duplicate, I do apologize.

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IIRC the $L^p$ norms with $p = 3, 4$ and a few other strange values occur in various inequalities, maybe in PDEs or harmonic analysis or something like that. I don't know the details though. (These are the infinite-dimensional versions of the $\ell^p$ norm on $\mathbb{R}^n$: see en.wikipedia.org/wiki/Lp_space for details.) –  Qiaochu Yuan Aug 29 '13 at 18:02
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One would naturally use the $\lVert\,\cdot\,\rVert_p$ norm when dealing with vector-valued $L^p$ functions. –  Daniel Fischer Aug 29 '13 at 18:05
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3 Answers

Think of $\mathbb R^2$ and $\mathbb R^3$, and picture the unit balls corresponding to the various norms. For the 1, 2, and $\infty$ norms in $\mathbb R^2$, you get a diamond, a circle, and a square, respectively. Such shapes are commonplace and "natural" in some sense, so these norms provide rather conventional ways of measuring distances.

For other values of $k$, the unit ball has some strange curved shape that is not likely to correspond with any physical measurement. I would say that this is why these norms are not used much in practice.

There is one use that I know of: in geometric modeling and computer graphics, people sometimes use objects of the form $\Vert \mathbf x - \mathbf a\Vert_k \le r$, where $k > 2$, to model shapes. Objects like these are sometimes called hyperellipsoids or superellipses. They are useful because adjusting $k$ lets you produce various different pleasingly smooth shapes. Also, their equations are not too complicated, so they can be handled in computations like ray-tracing. Specifically, it's fairly easy to decide whether a given point is inside or outside the shape.

Here is a link, and here is another one.

As the Wikipedia article points out, superellipses are sometimes used in font design (or, at least, Bezier curve approximations of superellipses are used).

See also squircles.

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I liked a lot the use Wikipedia mentions: those norms are useful in the making of dinner plates! :-) That's a practical use for sure. –  Giuseppe Negro Aug 30 '13 at 11:18
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Use of the $l^3$ norm in a problem would be quite unusual. However, use of the $l^p$ norm, where $p$ ranges over $(1,+\infty)$ is not at all unusual.

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How about the $L^1$ norm, or "Taxicab" norm. As the alternative name implies, it is the travel distance in a cab between two places on the road in New York.

(Because the road system New York can be approximated as criss-cross of vertical and horizontal straight lines.)

Also the $L^\infty$ norm is the maximum separation across all the coordinates between two points.

Note also that $L^2$ is the only norm which is rotation invariant, all other $L^p$ norms will change if you rotate the system, which renders nearly all of them undesirable for most purposes.

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I wanted to know about norms other than the 1,2 and $\infty$-norms, as stated in my first sentence. If you could omit all but the last paragraph and expand on that, that would be awesome. Also, I'm less interested in why we don't want to use them, than where they are actually useful. –  Jeremy Aug 29 '13 at 18:42
    
Ahh. Man. I meant rotation invariant. It is so because the map $(x,y)\to (x\cos(\theta)+y\sin(\theta),-x\sin(\theta)+y\cos(\theta))$ will preserve distance between any $(x_1,y_1)$, $(x_2,y_2)$, only with the $L^2$ norm. –  KalEl Aug 30 '13 at 2:02
    
@KalEl -- and that's because spheres and circles are the only unit ball shapes that are radially symmetric. –  bubba Aug 30 '13 at 10:32
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