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.

is a subset of a real or complex vector space then the or is defined to be the function valued in the extended real numbers where the infimum of the empty set is defined to be positive infinity.

is often assumed to have properties, such as being an absorbing disk will be a seminorm satisfying

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 into certain

). This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals is not empty is usually assumed to have properties (such as being absorbing is real-valued.

Definition

Define the or the valued in the extended real numberss, defined by

(recall that the infimum

can be extended remain undefined.

Some conditions making a gauge real-valued

In the field of convex analysis

to belong to the or is absorbing then the origin belongs to the algebraic interior is real-valued are given below.

Motivating examples

Example 1

Consider a normed vector space

Example 2

be any linear functional has the following properties:

1. It is

2. It is

3. It is

is a seminorm 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.

Consequently, the resulting topology need not be Hausdorff.

Common conditions guaranteeing gauges are seminorms

to be a disk

is convex and the origin belongs to the algebraic interior is a nonnegative sublinear functional which implies in particular that it is subadditive and positive homogeneous are balanced.

Gauges of absorbing disks

be an absorbing disk they can be applied in this special case.

Convexity and subadditivity

is obtained after the obvious modification.

is absorbing.

Balancedness and absolute homogeneity

Algebraic properties

• is a seminorm

• is a norm does not contain a non-trivial vector subspace.

• is absorbing

Topological properties

is a (real or complex) topological vector space (not necessarily Hausdorff or locally convex Then

is the topological interior is the topological closure Importantly, it was had any topological properties.

is continuous then

Minimal requirements on the set

This section will investigate the most general case of the gauge of is assumed to be an absorbing disk was discussed above.

Properties

is an absorbing disk.

The proofs of these basic properties are straightforward exercises so only the proofs of the most important statements are given.

is necessarily absorbing is straightforward and can be found in the article on absorbing sets.

so that taking the infimum of both sides shows that

and all is real-valued.

Examples

1. satisfies

could be proper.

Example

Example

Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are that have a certain purely algebraic property that is commonly encountered.

has not yet been ruled out).

-valued maps (for example, real-valued sublinear functions (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

Characterizing Minkowski functionals on star sets

Characterizing Minkowski functionals that are seminorms

is is balanced be convex.

Positive sublinear functions and Minkowski functionals

It may be shown that a real-valued subadditive function on an arbitrary topological vector space is continuous.

A is a nonnegative homogeneous is convex.

Correspondence between open convex sets and positive continuous sublinear functions

See also

Notes

References

Further reading

• 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). .