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
This is the Minkowski functional of K.[1] Usually it is assumed that K is such that the set of 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
One can see that , 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
Again we define
Then
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
by
which is often called the gauge of .
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.
- The set K being convex implies the subadditivity of pK.
- 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
But the left hand side is ½ pK(x + y), i.e. the above becomes
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
Therefore
See also
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.