What are some uses for other norms on $\mathbb{R}^n$ 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. 
 A: 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.
A: 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.
A: 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.
