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

$p(x)=\inf\{\lambda \in \mathbb {R} _{>0}:x\in \lambda K\}$ This is the Minkowski functional of K. Usually it is assumed that K is such that the set of $\lambda$ 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 $X$ , with the norm ||·||. Let $K$ be the unit ball in $X$ . Define a function $p:X\rightarrow \mathbb {R}$ by

$p(x)=\inf \left\{r>0:x\in rK\right\}.$ One can see that $p(x)=\|x\|$ , i.e. $p$ is just the norm on $X$ . 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 $\mathbb {K}$ . Take $\phi \in X'$ , the algebraic dual of $X$ , i.e. $\phi :X\rightarrow \mathbb {K}$ is a linear functional on $X$ . Fix $a>0$ . Let the set $K$ be given by

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

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

$p(x)={\frac {1}{a}}|\phi (x)|.$ The function p(x) is another instance of a Minkowski functional. It has the following properties:

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

Therefore, $p$ is a seminorm on $X$ , 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, $p(x)=0$ need not imply $x=0$ . In the above example, one can take a nonzero $x$ from the kernel of $\phi$ . 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

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

$p_{K}(x)=\inf \left\{r>0:x\in rK\right\},$ which is often called the gauge of $K$ .

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

$p_{K}\left({\frac {1}{2}}x+{\frac {1}{2}}y\right)\leq r+\epsilon ={\frac {1}{2}}p_{K}(x)+{\frac {1}{2}}p_{K}(y)+\epsilon .$ But the left hand side is ½ pK(x + y), i.e. the above becomes

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

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

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