# What does the graph of a norm look like

I talking about functions of the form ||X||^p, when p>1 for different values of p.

I know these are all convex functions, but I don't know how to graph them.

• See the wikipedia page on the superellipse: en.wikipedia.org/wiki/Superellipse – Ethan Bolker May 17 '16 at 20:45
• I don't see how the function in the link is related to the norm. – Erock Brox May 17 '16 at 21:59
• It may not be. I though you might be asking about the $p$-norm. – Ethan Bolker May 18 '16 at 0:03

Let $f\colon A \to B$ be a function. This means $f$ takes as an argument elements of the set $A$ and returns elements of the set $B$.

For example, $g(x) = x^2$ usually is a function that takes and returns real numbers, thus $g\colon \mathbb R\to\mathbb R$.

Usually the graph of a function $f$ can only be sketched, if the sum of the dimensions of $A$ and $B$ is less or equal to 3.

In order to graph functions that map from $\mathbb R$ to $\mathbb R$ you can use e.g. desmos.

Now you were originally asking about a function $\|X\|^p$. Your notation implies that $X$ is supposed to be a vector. As I said before you can only graph a functions if the sum of the dimensions is less or equal to 3. This means you can graph $\|X\|^p$, if $X$ is two-dimensional. You can use this in order to graph it in this case. Note that it cannot handle the norm directly. The usual norms for two dimesional vectors are \begin{align*} \left\|\begin{bmatrix}x\\y\end{bmatrix}\right\|_q &= \left(|x|^q + |y|^q\right)^{\frac1q} \end{align*} for some $0<q<\infty$. You can thus graph it by typing

( (abs(x)^1.5 + abs(y)^1.5)^(1/1.5) )^5

into the graphic 3D plotter from above. It will give you a plot of $\|X\|_q^p$, where $q=1.5$ and $p=5$.