Laplace expansion

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 convex body in a linear space V. We define a function p on V as

$p(x)=\inf \{\lambda \in {\mathbb{ℝ}}_{>0}:x\in \lambda K\}$

if that infimum is well-defined.

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

$p(x)=\inf \{r>0:x\in rK\}.$

One can see that $p(x)=\Vert x\Vert$, 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

$K=\{x\in X:|\varphi (x)|\le a\}.$

Again we define

$p(x)=\inf \{r>0:x\in rK\}.$

Then

$p(x)=\frac{1}{a}|\varphi (x)|.$

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

$p_K:X\to [0,\mathrm{\infty })$

by

$p_K(x)=\inf \{r>0:x\in rK\},$

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 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.

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

$p_K(\frac{1}{2}x+\frac{1}{2}y)\le r+ϵ=\frac{1}{2}{p}_K(x)+\frac{1}{2}{p}_K(y)+ϵ.$

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

$p_K(x+y)\le p_K(x)+p_K(y)+ϵ,\text{for all}ϵ>0.$

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

$\lambda x\in rK\text{if and only if}x\in \frac{r}{\left|\lambda \right|}K.$

Therefore

$p_K(\lambda x)=\inf \{r>0:\lambda x\in rK\}=\inf \{r>0:x\in \frac{r}{\left|\lambda \right|}K\}=\inf \left\{\right|\lambda |\frac{r}{\left|\lambda \right|}>0:x\in \frac{r}{\left|\lambda \right|}K\}=\left|\lambda \right|p_K(x).$

See also

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

Notes

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References

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