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, norm, topology, etc.) and the linear functions defi ...
, a Minkowski functional (after
Hermann Minkowski) 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
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 valued in the
extended real numbers, defined by
where the
infimum of the empty set is defined to be
positive infinity (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:
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.
This property of being nonnegative stands in contrast to other classes of functions, such as
sublinear functions 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 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 a ...
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 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 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
(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 a ...
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 (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 of
then
is a nonnegative
sublinear functional on
which implies in particular that it is
subadditive and
positive homogeneous
In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''de ...
.
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 (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 a ...
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 (real or complex)
topological vector space (TVS) (not necessarily
Hausdorff or
locally convex) 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 (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