Deriving a Norm From a Set
A few excellent propositions from Fleming's Functions of Several Variables:
Prop 1: Suppose $ \| \cdot \|$ is a norm on $ \mathbb{R}^n$. Then the unit ball about 0, $ \mathcal{B}:=\{ \mathbf{x} : \| \mathbf{x} \| \leq 1 \}$ has the following four properties:
It is compact.
It is convex.
$ \mathbf{x} \in \mathcal{B} \Rightarrow -\mathbf{x} \in \mathcal{B}$
$ \mathcal{B}$ contains a neighborhood of $ \mathbf{0}$ with respect to the normal metric on $ \mathbb{R}^n$.
Prop 2: Suppose $ K$ is a set with properties 1-4 above. Then the function $ \mathbb{R}^n \to \mathbb{R}$ given by
$$ \displaystyle \mathbf{0}\mapsto 0 $$
$$ \displaystyle \mathbf{x} \mapsto \frac{1}{\max \{t: t\mathbf{x} \in K\}}$$
is a norm.
Prop 3: The set $ \{\mathbf{x} \in \mathbb{R}^n : \sum_{i=1}^n |x_i|^p \leq 1\}$ has properties 1-4, and the derived norm as in prop 2 is the normal $ p$-norm.
So we have derived Minkowski's inequality indirectly.
