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.^[1] 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 → 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 K. Take φ ∈ X' , the algebraic dual of X, i.e. φ : X → 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:

It is subadditive: p(x + y) ≤ p(x) + p(y),
It is homogeneous: for all α ∈ K, p(α x) = |α| p(x),
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 φ. 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 )

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: x ∈ r K} is nonempty for every x. In order for p_K to have the properties of a seminorm, additional restrictions must be imposed on K. These conditions are listed below.

The set K being convex implies the subadditivity of p_K.
Homogeneity, i.e. p_K(α x) = |α| p_K(x) for all α, is ensured if K is balanced, meaning α K ⊂ K 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 p_K(x) = p_K(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 p_K, 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 p_K(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 p_K(x) > p_K(y) is obtained after the obvious modification.

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

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).

Notes

↑ Thompson (1996) p.17

References

Thompson, Anthony C. (1996). Minkowski Geometry. Encyclopedia of Mathematics and Its Applications. Cambridge University Press. ISBN 0-521-40472-X.

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