Minkowski functional

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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

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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

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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

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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

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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

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

  •   is a seminorm on  
  •   is a norm on   if and only if   does not contain a non-trivial vector subspace.[4]
  •   for any scalar  [4]
  • 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  [5]
    • In particular, if   is as above and   is any seminorm on   then   if and only if  [5]
  • If   satisfies   then  

Topological properties

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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

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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

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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

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  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

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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

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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

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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

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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

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Notes

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  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

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  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

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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