In mathematics, in the field of functional analysis, 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 then the Minkowski functional or gauge 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 not a real number so that would then not be realvalued).
The set is often assumed/picked to have properties, such as being an absorbing disk in that guarantee that will be a realvalued 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 (nonuniquely) 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 nonnegative (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 realvalued 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 realvalued.
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 nonzero real The products and remain undefined.
Some conditions making a gauge realvalued
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 realvalued (that is, to never take on the value of ), which happens if and only if the set is nonempty for every
In order for to be realvalued, 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 realvalued. Characterizations of when is realvalued are given below.
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:
Therefore, is a seminorm on with an induced topology. This is characteristic of Minkowski functionals defined via "nice" sets. There is a onetoone 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.
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 absorbing disk 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 realvalued function on that is positive homogeneous, then the sets and satisfy and if in addition is absolutely homogeneous then both and are balanced.^{[3]}
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.
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 
Let be a real or complex vector space and let be an absorbing disk in
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 ^{[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]}
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 absorbing disk in was discussed above.
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
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 welldefined, it is necessary and sufficient that thus for all and all nonnegative real if and only if is realvalued. 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 realvalued. 
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 nonzero 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:
Moreover, if never takes on the value (so that the product is always welldefined) then this list may be extended to include:
Proof


If holds for all and real then so that 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, realvalued 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.
Proposition^{[10]} — Let be any function and be any subset. The following statements are equivalent:
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:
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
It may be shown that a realvalued 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 realvalued 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 nonempty 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. 