Minkowski functional

From Wikipedia, the free encyclopedia

This article does not cite any references or sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2008)

In functional analysis, given a linear space X, a Minkowski functional is a device that uses the linear structure to introduce a topology on X.

Contents 1 Motivation 1.1 Example 1 1.2 Example 2 2 Definition 2.1 Convexity of K 2.2 Balancedness of K 3 Related links

[edit] Motivation

[edit] Example 1

Consider a normed vector space X, with the norm ||·||. Let K be the unit sphere in X. Define a function p : X → R by

One can see that , i.e. p is just the norm on X. The function p is a special case of a Minkowski functional.

[edit] 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

Again we define

Then

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's 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.

[edit] 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

by

It is implicitly assumed in this definition that 0 ∈ K and the set {r > 0: x ∈ r K} is nonempty. In order for p_K 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 p_K.
2. 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.

[edit] 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

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

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.

[edit] Balancedness of K

Notice that K being balanced implies that

Therefore

[edit] Related links

• Hadwiger's theorem
• Hugo Hadwiger
• Morphological image processing