where the infimum of the empty set is defined to be positive infinity (which is not a real number so that would then not be real-valued).
The set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a real-valued seminorm on
In fact, every seminorm on is equal to the Minkowski functional (that is, ) of any subset of satisfying (where all three of these sets are necessarily absorbing in and the first and last are also disks).
Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can be associated with its Minkowski functional (which will necessarily be a seminorm).
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 geometric properties of a subset of into certain algebraic properties of a function on
The Minkowski function is always non-negative (meaning ).
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 (such as being absorbing in for instance) that will guarantee that is real-valued.
Definitionedit
Let be a subset of a real or complex vector space Define the gauge of or the Minkowski functional associated with or induced by as being the function valued in the extended real numbers, defined by
where recall that the infimum of the empty set is (that is, ). 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 (that is, to never take on the value of ), 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 algebraic interior or core of in [1]
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 examplesedit
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 (not necessarily continuous).
Fix
Let be the set
and let be the Minkowski functional of
Then
The function has the following properties:
It is subadditive:
It is absolutely homogeneous: for all scalars
It is nonnegative:
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 seminormsedit
To guarantee that it will henceforth be assumed that
In order for to be a seminorm, it suffices for to be a disk (that is, convex and balanced) and absorbing in which are the most common assumption placed on
Theorem[2] — If is an absorbingdisk in a vector space then the Minkowski functional of which is the map defined by
is a seminorm on
Moreover,
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 [3]
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.[3]
Gauges of absorbing disksedit
Arguably the most common requirements placed on a set to guarantee that is a seminorm are that be an absorbingdisk 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 (if any) assumptions on they can be applied in this special case.
Theorem — Assume that is an absorbing subset of
It is shown that:
Proof that the Gauge of an absorbing disk is a seminorm
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 propertiesedit
Let be a real or complex vector space and let be an absorbing disk in
where is the topological interior and is the topological closure of in [6]
Importantly, it was not 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 [6]
If is continuous then[6]
Minimal requirements on the setedit
This section will investigate the most general case of the gauge of any subset of
The more common special case where is assumed to be an absorbingdisk in was discussed above.
Propertiesedit
All results in this section may be applied to the case where is an absorbing disk.
Throughout, is any subset of
Summary — Suppose that is a subset of a real or complex vector space
A map is called nonnegative homogeneous[7] if for all and all nonnegative real Since is undefined, a map that takes infinity as a value is not nonnegative homogeneous.
Real-values: is the set of all points on which is real valued. So is real-valued if and only if in which case
Value at : if and only if if and only if
Null space: If then if and only if if and only if there exists a divergent sequence of positive real numbers such that for all Moreover, the zero set of is
Comparison to a constant: If then for any if and only if this can be restated as: If then
It follows that if is real then where the set on the right hand side denotes and not its subset If then these sets are equal if and only if contains
In particular, if or then but importantly, the converse is not necessarily true.
Gauge comparison: For any subset if and only if thus if and only if
The assignment is order-reversing in the sense that if then [8]
Because the set satisfies it follows that replacing with will not change the resulting Minkowski functional. The same is true of and of
If then and has the particularly nice property that if is real then if and only if or [note 1] Moreover, if is real then if and only if
Subadditive/Triangle inequality: is subadditive if and only if is convex. If is convex then so are both and and moreover, is subadditive.
Scaling the set: If is a scalar then for all
Thus if is real then
Symmetric: is symmetric (meaning that for all ) if and only if is a symmetric set (meaning that), which happens if and only if
Absolute homogeneity: for all and all unit length scalars [note 2] if and only if for all unit length scalars in which case for all and all non-zero scalars If in addition is also real-valued then this holds for all scalars (that is, is absolutely homogeneous[note 3]).
for all unit length if and only if for all unit length
for all unit scalars if and only if for all unit scalars if this is the case then for all unit scalars
Absorbing: If is convex or balanced and if then is absorbing in
If a set is absorbing in and then is absorbing in
If is convex and then in which case
Restriction to a vector subspace: If is a vector subspace of and if denotes the Minkowski functional of on then where denotes the restriction of to
Proof
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 non-negative real if and only if is real-valued.
The hypothesis of statement (7) 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.
Examplesedit
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.
The next theorem shows that Minkowski functionals are exactly those functions that have a certain purely algebraic property that is commonly encountered.
Theorem — Let be any function.
The following statements are equivalent:
Only (1) implies (3) 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 (since ), 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 (importantly, however, the possibility that is or has not yet been ruled out).
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 (so in particular, is guaranteed).
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 (for example, real-valued sublinear functions) in terms of Minkowski functionals.
For instance, it can be used to describe how every real homogeneous function (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.
Characterizing Minkowski functionals on star setsedit
Proposition[10] — Let be any function and be any subset.
The following statements are equivalent:
is (strictly) positive homogeneous, and
is the Minkowski functional of (that is, ), contains the origin, and is star-shaped at the origin.
The set is star-shaped at the origin if and only if whenever and A set that is star-shaped at the origin is sometimes called a star set.[9]
Characterizing Minkowski functionals that are seminormsedit
In this next theorem, which follows immediately from the statements above, is not 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 (which is a property that is often required to have); 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.
Theorem — Let be a subset of a real or complex vector space
Then is a seminorm on if and only if all of the following conditions hold:
It suffices (but is not necessary) for to be convex.
for all unit scalars
This condition is satisfied if is balanced or more generally if for all unit scalars
in which case and both and will be convex, balanced, and absorbing subsets of
Conversely, if is a seminorm on then the set satisfies all three of the above conditions (and thus also the conclusions) and also
moreover, is necessarily convex, balanced, absorbing, and satisfies
Corollary — If is a convex, balanced, and absorbing subset of a real or complex vector space then is a seminorm on
Positive sublinear functions and Minkowski functionalsedit
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 [11]
If is subadditive and satisfies then is continuous if and only if its absolute value is continuous.
A nonnegative sublinear function 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
Theorem[11] — Suppose that is a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers.
Then the non-empty open convex subsets of are exactly those sets that are of the form for some and some positive continuous sublinear function on
Proof
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 (however, is not necessarily a seminorm since it is not necessarily absolutely homogeneous).
From the properties of Minkowski functionals, we have from which it follows that and so
Since this completes the proof.
Morphological image processing – theory and technique for the analysis and processing of geometrical structuresPages displaying wikidata descriptions as a fallback
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