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.

Motivation

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

p(x) = \inf \left\{r > 0: x \in r K \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 r K \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) ≤ 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 \rightarrow [0, \infty)

by

p_K (x) = \inf \left\{r > 0: x \in r K \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. 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\left( \frac{1}{2} x + \frac{1}{2} y\right) \le 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) \le 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: x ∈ r K} is nonempty, implies that K is absorbent.

Balancedness of K

Notice that K being balanced implies that

\lambda x \in r K \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 r K \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).

See also

Notes

  1. ↑ Thompson (1996) p.17

References