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 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.

Definition

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 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:

  1. It is subadditive:
  2. It is absolutely homogeneous: for all scalars
  3. 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 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

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 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 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.

Theorem — Assume that is an absorbing subset of It is shown that:

  1. If is convex then is subadditive.
  2. If is balanced then is absolutely homogeneous; that is, for all scalars
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 properties

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

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 [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 set

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.

Properties

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

  1. Strict positive homogeneity: for all and all positive real
    • Positive/Nonnegative homogeneity: is nonnegative homogeneous if and only if is real-valued.
      • 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.
  2. 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
  3. 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.
  4. 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
  5. Subadditive/Triangle inequality: is subadditive if and only if is convex. If is convex then so are both and and moreover, is subadditive.
  6. Scaling the set: If is a scalar then for all Thus if is real then
  7. Symmetric: is symmetric (meaning that for all ) if and only if is a symmetric set (meaning that), which happens if and only if
  8. 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
    • The Minkowski functional of any balanced set is a balanced function.[8]
  9. 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
  10. 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.

Examples

  1. If is a non-empty collection of subsets of then for all where
    • Thus for all
  2. 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 exactly those functions that have a certain purely algebraic property that is commonly encountered.

Theorem — Let be any function. The following statements are equivalent:

  1. Strict positive homogeneity: for all and all positive real
    • This statement is equivalent to: for all and all positive real
  2. is a Minkowski functional: meaning that there exists a subset such that
  3. where
  4. where

Moreover, if never takes on the value (so that the product is always well-defined) then this list may be extended to include:

  1. Positive/Nonnegative homogeneity: for all and all nonnegative real
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, 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 sets

Proposition[10] — Let be any function and be any subset. The following statements are equivalent:

  1. is (strictly) positive homogeneous, and
  2. 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 seminorms

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:

  1. (or equivalently, is real-valued).
  2. is convex (or equivalently, is subadditive).
    • It suffices (but is not necessary) for to be convex.
  3. 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 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 [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.

See also

Notes

  1. ^ It is in general false that if and only if (for example, consider when is a norm or a seminorm). The correct statement is: If then if and only if or
  2. ^ is having unit length means that
  3. ^ The map is called absolutely homogeneous if is well-defined and for all and all scalars (not just non-zero scalars).

References

  1. ^ Narici & Beckenstein 2011, p. 109.
  2. ^ Narici & Beckenstein 2011, p. 119.
  3. ^ a b Jarchow 1981, pp. 104–108.
  4. ^ a b Narici & Beckenstein 2011, pp. 115–154.
  5. ^ a b Schaefer 1999, p. 40.
  6. ^ a b c Narici & Beckenstein 2011, p. 119-120.
  7. ^ Kubrusly 2011, p. 200.
  8. ^ a b Schechter 1996, p. 316.
  9. ^ Schechter 1996, p. 303.
  10. ^ Schechter 1996, pp. 313–317.
  11. ^ a b Narici & Beckenstein 2011, pp. 192–193.

Further reading