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Minkowski functional

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

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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 { λ ∈ R > 0 : x ∈ λ K }

This is the Minkowski functional of K. Usually it is assumed that K is such that the set of λ is never empty, but sometimes the set is allowed to be empty and then p(x) is defined as infinity.

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 { r > 0 : x ∈ r K } .

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 ∈ X : | ϕ ( x ) | ≤ a } .

Again we define

p ( x ) = inf { r > 0 : x ∈ r K } .

Then

p ( x ) = 1 a | ϕ ( x ) | .

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

  1. It is subadditive: p(x + y) ≤ p(x) + p(y),
  2. It is homogeneous: for all α ∈ K, p(α x) = |α| 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 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 → [ 0 , ∞ )

by

p K ( x ) = inf { r > 0 : x ∈ r K } ,

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 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 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 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 ( 1 2 x + 1 2 y ) ≤ r + ϵ = 1 2 p K ( x ) + 1 2 p K ( y ) + ϵ .

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

p K ( x + y ) ≤ p K ( x ) + p K ( y ) + ϵ , for all ϵ > 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: x ∈ r K} is nonempty, implies that K is absorbent.

Balancedness of K

Notice that K being balanced implies that

λ x ∈ r K if and only if x ∈ r | λ | K .

Therefore

p K ( λ x ) = inf { r > 0 : λ x ∈ r K } = inf { r > 0 : x ∈ r | λ | K } = inf { | λ | r | λ | > 0 : x ∈ r | λ | K } = | λ | p K ( x ) .

References

Minkowski functional Wikipedia
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