Minkowski functional

is a subset of a real or complex vector space

valued in the extended real numbers, defined by

where the infimum of the empty set is defined to be positive infinity

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).

In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of

be a subset of a real or complex vector space

valued in the extended real numbers, defined by

be a vector space without topology with underlying scalar field

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,

is convex and the origin belongs to the algebraic interior of

[3] Arguably the most common requirements placed on a set

Due to how common these assumptions are, the properties of a Minkowski functional

Since all of the results mentioned above made few (if any) assumptions on

is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let

This section will investigate the most general case of the gauge of any subset

is a subset of a real or complex vector space

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

This proves that Minkowski functionals are strictly positive homogeneous.

The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of

that have a certain purely algebraic property that is commonly encountered.

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.

For instance, it can be used to describe how every real homogeneous function

be a subset of a real or complex vector space

is necessarily convex, balanced, absorbing, and satisfies

is a convex, balanced, and absorbing subset of a real or complex vector space

is a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers.

where this convex open neighborhood of the origin satisfies