Minkowski functional

In mathematics, in the field of functional analysis, a Minkowski functional or gauge function is a function that recovers a notion of distance on a linear space.
If is a subset of a real or complex vector space then the or of is defined to be the function valued in the Extended [real number line|extended real number]s, defined by
where the infimum of the empty set is defined to be positive infinity.
The set is often assumed to have properties, such as being an absorbing disk in, that guarantee that will be a seminorm on In fact, every seminorm on is equal to the Minkowski functional of any subset of satisfying
.
Thus every seminorm can be associated with an absorbing disk and conversely, every absorbing disk can be associated with its Minkowski functional.
These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis.
In particular, through these relationships, Minkowski functionals allow one to "translate" certain properties of a subset of into certain properties of a function on
The Minkowski function is always non-negative.
This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals, that do allow negative values.
However, might not be real-valued since for any given the value is a real number if and only if is not empty.
Consequently, is usually assumed to have properties that will guarantee that is real-valued.

Definition

Let be a subset of a real or complex vector space Define the of or the associated with or induced by as being the function valued in the extended real numbers, defined by
. Here, is shorthand for
For any if and only if is not empty.
The arithmetic operations on can be extended to operate on where for all non-zero real
The products and remain undefined.

Some conditions making a gauge real-valued

In the field of convex analysis, the map taking on the value of is not necessarily an issue.
However, in functional analysis is almost always real-valued, which happens if and only if the set is non-empty for every
In order for to be real-valued, it suffices for the origin of to belong to the or of in
If is absorbing in where recall that this implies that then the origin belongs to the algebraic interior of in and thus is real-valued.
Characterizations of when is real-valued are given below.

Motivating examples

Example 1

Consider a normed vector space with the norm and let be the unit ball in Then for every Thus the Minkowski functional is just the norm on

Example 2

Let be a vector space without topology with underlying scalar field
Let be any linear functional on .
Fix
Let be the set
and let be the Minkowski functional of
Then
The function has the following properties:

It is :
It is : for all scalars
It is :

Therefore, is a seminorm on 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, need not imply
In the above example, one can take a nonzero from the kernel of
Consequently, the resulting topology need not be Hausdorff.

Common conditions guaranteeing gauges are seminorms

To guarantee that it will henceforth be assumed that
In order for to be a seminorm, it suffices for to be a disk and absorbing in which are the most common assumption placed on
More generally, if is convex and the origin belongs to the algebraic interior of then is a nonnegative sublinear functional on which implies in particular that it is subadditive and positive homogeneous.
If is absorbing in then is positive homogeneous, meaning that for all real where
If is a nonnegative real-valued function on that is positive homogeneous, then the sets and satisfy and
if in addition is absolutely homogeneous then both and are balanced.

Gauges of absorbing disks

Arguably the most common requirements placed on a set to guarantee that is a seminorm are that be an absorbing disk in
Due to how common these assumptions are, the properties of a Minkowski functional when is an absorbing disk will now be investigated.
Since all of the results mentioned above made few assumptions on they can be applied in this special case.
Convexity and subadditivity
A simple geometric argument that shows convexity of implies subadditivity is as follows.
Suppose for the moment that
Then for all
Since is convex and is also convex.
Therefore,
By definition of the Minkowski functional
But the left hand side is so that
Since was arbitrary, it follows that which is the desired inequality.
The general case is obtained after the obvious modification.
Convexity of together with the initial assumption that the set is nonempty, implies that is absorbing.
Balancedness and absolute homogeneity
Notice that being balanced implies that
Therefore

Algebraic properties

Let be a real or complex vector space and let be an absorbing disk in

is a seminorm on
is a norm on if and only if does not contain a non-trivial vector subspace.
for any scalar
If is an absorbing disk in and then
If is a set satisfying then is absorbing in and where is the Minkowski functional associated with that is, it is the gauge of
In particular, if is as above and is any seminorm on then if and only if
If satisfies then

Topological properties

Assume that is a topological vector space and let be an absorbing disk in Then
where is the topological interior and is the topological closure of in
Importantly, it was assumed that was continuous nor was it assumed that had any topological properties.
Moreover, the Minkowski functional is continuous if and only if is a neighborhood of the origin in
If is continuous then

Minimal requirements on the set

This section will investigate the most general case of the gauge of subset of
The more common special case where is assumed to be an absorbing disk in was discussed above.

Properties

All results in this section may be applied to the case where is an absorbing disk.
Throughout, is any subset of
The proofs of these basic properties are straightforward exercises so only the proofs of the most important statements are given.
The proof that a convex subset that satisfies is necessarily absorbing in is straightforward and can be found in the article on absorbing sets.
For any real
so that taking the infimum of both sides shows that
This proves that Minkowski functionals are strictly positive homogeneous. For to be well-defined, it is necessary and sufficient that thus for all and all real if and only if is real-valued.
The hypothesis of statement allows us to conclude that for all and all scalars satisfying
Every scalar is of the form for some real where and is real if and only if is real.
The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of and from the positive homogeneity of when is real-valued.

Examples

If is a non-empty collection of subsets of then for all where
* Thus for all
If is a non-empty collection of subsets of and satisfies

then for all
The following examples show that the containment could be proper.
Example: If and then but which shows that its possible for to be a proper subset of when
The next example shows that the containment can be proper when the example may be generalized to any real
Assuming that the following example is representative of how it happens that satisfies but
Example: Let be non-zero and let so that and
From it follows that
That follows from observing that for every which contains
Thus and
However, so that as desired.

Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are those functions that have a certain purely algebraic property that is commonly encountered.
If holds for all and real then so that
Only implies will be proven because afterwards, the rest of the theorem follows immediately from the basic properties of Minkowski functionals described earlier; properties that will henceforth be used without comment.
So assume that is a function such that for all and all real and let
For all real so by taking for instance, it follows that either or
Let
It remains to show that
It will now be shown that if or then so that in particular, it will follow that
So suppose that or in either case for all real
Now if then this implies that that for all real, which implies that as desired.
Similarly, if then for all real which implies that as desired.
Thus, it will henceforth be assumed that a positive real number and that .
Recall that just like the function satisfies for all real
Since if and only if so assume without loss of generality that and it remains to show that
Since which implies that .
It remains to show that which recall happens if and only if
So assume for the sake of contradiction that and let and be such that where note that implies that
Then
This theorem can be extended to characterize certain classes of -valued maps in terms of Minkowski functionals.
For instance, it can be used to describe how every real homogeneous function can be written in terms of a unique Minkowski functional having a certain property.

Characterizing Minkowski functionals that are seminorms

In this next theorem, which follows immediately from the statements above, is assumed to be absorbing in and instead, it is deduced that is absorbing when is a seminorm. It is also not assumed that is balanced ; in its place is the weaker condition that for all scalars satisfying
The common requirement that be convex is also weakened to only requiring that be convex.

Positive sublinear functions and Minkowski functionals

It may be shown that a real-valued subadditive function on an arbitrary topological vector space is continuous at the origin if and only if it is uniformly continuous, where if in addition is nonnegative, then is continuous if and only if is an open neighborhood in
If is subadditive and satisfies then is continuous if and only if its absolute value is continuous.
A is a nonnegative homogeneous function that satisfies the triangle inequality.
It follows immediately from the results below that for such a function if then
Given the Minkowski functional is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if and is convex.

Correspondence between open convex sets and positive continuous sublinear functions

Let be an open convex subset of
If then let and otherwise let be arbitrary.
Let be the Minkowski functional of where this convex open neighborhood of the origin satisfies
Then is a continuous sublinear function on since is convex, absorbing, and open.
From the properties of Minkowski functionals, we have from which it follows that and so
Since this completes the proof.