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We defined a new norm. The norm of $x \in \mathbb{R}^n$ is defined as $$ N_P(x) = \min \{t \geq 0 : x \in t\cdot P\} \enspace,$$ where $P$ is a centrally symmetric and convex body centered at the origin point.

We prove that it is a norm.

1.Identity of indiscernibles.
Obviously, $N_P(x)=0 \Leftrightarrow x=0$.

2.Absolutely scalable.

Because of centrally symmetric property, $N_P(ax)=|a|N_P(x)$.

3.Triangle inequality.

Denote $N_P(x+y), N_P(x),N_P(y)$ as $t_0,t_1,t_2$ respectively. And let vectors $x+y,x,y$ go from the origin point and hit the border of $P$ at $a,b,c$ respectively.

Therefore $(x+y)=x+y$ implies $t_0\vec{a}=t_1\vec{b}+t_2\vec{c}$, implies $\vec{a}=\frac{t_1}{t_0}\vec{b}+\frac{t_2}{t_0}\vec{c}$

Suppose $t_0 > t_1+t_2$, thus $0\leq \frac{t_1}{t_0}+\frac{t_2}{t_0}<1$.

However, this contradict to the convex property because border $bac$ is not convex. QED

We realized this new norm consists all possible norms in $\mathbb{R}^n$, including $\ell_p$.

Because for any norm $N(\cdot)$, define $P=\{ x : N(x)\leq 1\}$, one can verify that $N_P=N$. It shows a simple fact: a norm is equivalent to the space which has unit norm.

Our question is, did anyone discover it before? What is the name? We googled it but did not get answers.

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    $\begingroup$ As the answer states, you are looking at a particular class of Minkowski functionals. But note that your conditions on $P$ are not sufficient to get a norm. Consider the case $n = 2$ and $P = \{ -1 < x < 1\}$. This set is convex and centrally symmetric, but $N_p((0,1)) = 0$. Similarly, you can consider the case where $P = \{ x = 0, |y| < 1\}$. In this case $N_p((1,0) = \infty$. Finally, if you allo your $P$ to be open (as I did in the first example), you really want to use $\inf$ instead of $\min$. $\endgroup$
    – Willie Wong
    Apr 12, 2021 at 13:24
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    $\begingroup$ In convex geometry this is the reciprocal of the radial function of a convex body. $\endgroup$
    – Deane Yang
    Apr 12, 2021 at 15:30

1 Answer 1

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These are called Minkowski functionals, see Wikipedia article. They come up in the study of locally convex vector spaces.

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