![Minkowski functional](https://upload.wikimedia.org/wikipedia/commons/a/ae/Minkowski_functional.svg)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in the field of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a Minkowski functional (after
Hermann Minkowski
Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
) or gauge function is a function that recovers a notion of distance on a linear space.
If
is a subset of a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
or
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
then the or of
is defined to be the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
valued in the
extended real number
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
s, defined by
where the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
of the empty set is defined to be
positive infinity
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
(which is a real number so that
would then be real-valued).
The Minkowski function is always non-negative (meaning
) and
is a real number if and only if
is not
empty
Empty may refer to:
Music Albums
* ''Empty'' (God Lives Underwater album) or the title song, 1995
* ''Empty'' (Nils Frahm album), 2020
* ''Empty'' (Tait album) or the title song, 2001
Songs
* "Empty" (The Click Five song), 2007
* ...
.
This property of being nonnegative stands in contrast to other classes of functions, such as
sublinear function In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
s and real
linear functionals
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the ...
, that do allow negative values.
In functional analysis,
is usually assumed to have properties (such as being
absorbing in
for instance) that will guarantee that for every
this set
is not empty precisely because this results in
being real-valued.
Moreover,
is also often assumed to have more properties, such as being an absorbing
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
in
since these properties guarantee that
will be a (real-valued)
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
on
In fact, every seminorm
on
is equal to the Minkowski functional 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 defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a 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 properties of a subset of
into certain properties of a function on
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 number
In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra ...
s, defined by
where recall that the
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
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
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex minimization, a subdomain of optimization theory.
Convex sets
A subset C \subseteq X of s ...
, 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 or of
in
If
is
absorbing in
where recall that this implies that
then the origin belongs to the
algebraic interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior.
Definition
Assume that A is a subset of a vector space X.
The ''algebraic i ...
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
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
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
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
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 :
#It is :
for all scalars
#It is :
Therefore,
is a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
(that is, convex and balanced) and absorbing in
which are the most common assumption placed on
More generally, if
is convex and the origin belongs to the
algebraic interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior.
Definition
Assume that A is a subset of a vector space X.
The ''algebraic i ...
of
then
is a nonnegative
sublinear functional In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. ...
on
which implies in particular that it is
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
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
In telecommunications and professional audio, a balanced line or balanced signal pair is a circuit consisting of two conductors of the same type, both of which have equal impedances along their lengths and equal impedances to ground and to other ...
.
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
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.
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 In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
on
- is a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envir ...
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 (real or complex)
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(TVS) (not necessarily
Hausdorff or
locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ...
) and let
be an absorbing disk in
Then
where
is the
topological interior and
is the
topological closure
In topology, the closure of a subset of points in a topological space consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary, and also as the intersection of ...
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
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 set
In functional analysis and related areas of mathematics an absorbing set in a vector space is a Set (mathematics), set S which can be "inflated" or "scaled up" to eventually always include any given point of the vector space.
Alternative terms are ...
s.
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 (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.
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
_is_a_Nonnegative_homogeneity.html" ;"title=", \infty) is continuous.
A positive
that satisfies the triangle inequality.
It follows immediately from the results below that for such a function
is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if
is convex.
Correspondence between open convex sets and positive continuous sublinear functions
Let
be arbitrary.
Let
is not necessarily a seminorm since it is not necessarily absolutely homogeneous).
From the properties of Minkowski functionals, it follows that
this completes the proof.
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* F. Simeski, A.M.P. Boelens and M. Ihme. Modeling Adsorption in Silica Pores via Minkowski Functionals and Molecular Electrostatic Moments. ''Energies'' 13 (22) 5976 (2020). https://doi.org/10.3390/en13225976
{{Convex analysis and variational analysis
Convex analysis
Functional analysis
Hermann Minkowski