# Minkowski functional

In mathematics, in the field of functional analysis, a Minkowski functional is a function that recovers a notion of distance on a linear space.

Let K be a symmetric (i.e. if it contains x it also contains -x) convex body in a linear space V. We define a function p on V as

${\displaystyle p(x)=\inf\{\lambda \in \mathbb {R} _{>0}:x\in \lambda K\}}$

This is the Minkowski functional of K.[1] Usually it is assumed that K is such that the set of ${\displaystyle \lambda }$ is never empty, but sometimes the set is allowed to be empty and then p(x) is defined as infinity.

## Examples

### Example 1

Consider a normed vector space ${\displaystyle X}$, with the norm ||·||. Let ${\displaystyle K}$ be the unit ball in ${\displaystyle X}$. Define a function ${\displaystyle p:X\rightarrow \mathbb {R} }$ by

${\displaystyle p(x)=\inf \left\{r>0:x\in rK\right\}.}$

One can see that ${\displaystyle p(x)=\|x\|}$, i.e. ${\displaystyle p}$ is just the norm on ${\displaystyle X}$. The function p is a special case of a Minkowski functional.

### Example 2

Let X be a vector space without topology with underlying scalar field ${\displaystyle \mathbb {K} }$. Take ${\displaystyle \phi \in X'}$, the algebraic dual of ${\displaystyle X}$, i.e. ${\displaystyle \phi :X\rightarrow \mathbb {K} }$ is a linear functional on ${\displaystyle X}$. Fix ${\displaystyle a>0}$. Let the set ${\displaystyle K}$ be given by

${\displaystyle K=\{x\in X:|\phi (x)|\leq a\}.}$

Again we define

${\displaystyle p(x)=\inf \left\{r>0:x\in rK\right\}.}$

Then

${\displaystyle p(x)={\frac {1}{a}}|\phi (x)|.}$

The function p(x) is another instance of a Minkowski functional. It has the following properties:

1. It is subadditive: ${\displaystyle p(x+y)\leqslant p(x)+p(y)}$,
2. It is homogeneous: for all ${\displaystyle \alpha \in \mathbb {K} ,p(\alpha x)=\left\vert \alpha \right\vert p(x)}$,
3. It is nonnegative.

Therefore, ${\displaystyle p}$ is a seminorm on ${\displaystyle X}$, with an induced topology. This is characteristic of Minkowski functionals defined via "nice" sets. There is a one-to-one correspondence between seminorms and the Minkowski functional given by such sets. What is meant precisely by "nice" is discussed in the section below.

Notice that, in contrast to a stronger requirement for a norm, ${\displaystyle p(x)=0}$ need not imply ${\displaystyle x=0}$. In the above example, one can take a nonzero ${\displaystyle x}$ from the kernel of ${\displaystyle \phi }$. Consequently, the resulting topology need not be Hausdorff.

## Definition

The above examples suggest that, given a (complex or real) vector space X and a subset K, one can define a corresponding Minkowski functional

${\displaystyle p_{K}:X\rightarrow [0,\infty )}$

by

${\displaystyle p_{K}(x)=\inf \left\{r>0:x\in rK\right\},}$

which is often called the gauge of ${\displaystyle K}$.

It is implicitly assumed in this definition that 0 ∈ K and the set {r > 0: xr K} is nonempty for every x. In order for pK to have the properties of a seminorm, additional restrictions must be imposed on K. These conditions are listed below.

1. The set K being convex implies the subadditivity of pK.
2. Homogeneity, i.e. pK(α x) = |α| pK(x) for all α, is ensured if K is balanced, meaning α KK for all |α| ≤ 1.

A set K with these properties is said to be absolutely convex.

### Convexity of K

A simple geometric argument that shows convexity of K implies subadditivity is as follows. Suppose for the moment that pK(x) = pK(y) = r. Then for all ε > 0, we have x, y ∈ (r + ε) K = K' . The assumption that K is convex means K' is also. Therefore, ½ x + ½ y is in K' . By definition of the Minkowski functional pK, one has

${\displaystyle p_{K}\left({\frac {1}{2}}x+{\frac {1}{2}}y\right)\leq r+\epsilon ={\frac {1}{2}}p_{K}(x)+{\frac {1}{2}}p_{K}(y)+\epsilon .}$

But the left hand side is ½ pK(x + y), i.e. the above becomes

${\displaystyle p_{K}(x+y)\leq p_{K}(x)+p_{K}(y)+\epsilon ,\quad {\mbox{for all}}\quad \epsilon >0.}$

This is the desired inequality. The general case pK(x) > pK(y) is obtained after the obvious modification.

Note Convexity of K, together with the initial assumption that the set {r > 0: xr K} is nonempty, implies that K is absorbing.

### Balancedness of K

Notice that K being balanced implies that

${\displaystyle \lambda x\in rK\quad {\mbox{if and only if}}\quad x\in {\frac {r}{|\lambda |}}K.}$

Therefore

${\displaystyle p_{K}(\lambda x)=\inf \left\{r>0:\lambda x\in rK\right\}=\inf \left\{r>0:x\in {\frac {r}{|\lambda |}}K\right\}=\inf \left\{|\lambda |{\frac {r}{|\lambda |}}>0:x\in {\frac {r}{|\lambda |}}K\right\}=|\lambda |p_{K}(x).}$