In mathematics, in the field of functional analysis, a Minkowski functional is a function that recovers a notion of distance on a linear space. Let be a symmetric convex body in a linear space. We define a function on as This is the Minkowski functional of. Usually it is assumed that is such that the set of is never empty, but sometimes the set is allowed to be empty and then is defined as infinity.
Definition
Given a real or complex vector space and an absorbing subset, one can 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 . ;Conditions making a Minkowski functional real-valued In the field of convex analysis, the map taking on the value of is not necessarily an issue. However, in functional analysis we almost alwayswant to be real-valued, which happens if and only if the set is non-empty for every. Note in particular, that if and only if, which we will henceforth assume to be the case. In order for to be real-valued, it suffices for the origin to belong to the algebraic interior of. If is absorbing in, where recall that implies, then the origin to belong to the algebraic interior of and thus is real-valued. ;Conditions making a Minkowski functional into a seminorm If is convex and the origin to belong to the algebraic interior of, then is a non-negative sublinear functional on, which implies in particular that it is subadditive and positive homogeneous. In order for to be a seminorm, it suffices for to be a disk and absorbing in, which are the most common assumption placed on.
Proving a that a Gauge is a seminorm
We assume that is an absorbing subset of. We show that:
A simple geometric argument that shows convexity of implies subadditivity is as follows. Suppose for the moment that. For all, we have. Since is convex and, is also convex. Therefore,. By definition of the Minkowski functional, we have 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. Note Convexity of, together with the initial assumption that the set is nonempty, implies that is absorbing.
Balancedness and homogeneity
Notice that being balanced implies that Therefore
Properties
Algebraic properties
Let be a real or complex vector space and let be an absorbing disk in.
Note that we did not assume that was continuous or that had any topological properties.
is continuous if and only if is a neighborhood of the origin in.
If is continuous then:
, and
.
Motivating examples
Example 1
Consider a normed vector space, with the norm ||·||. Let be the unit ball in. Define a function by One can see that, i.e. is just the norm on. The function is a special case of a Minkowski functional.
Example 2
Let be a vector space without topology with underlying scalar field. Take, the algebraic dual of, i.e. is a linear functional on. Fix. Let the set be given by Again we define Then The function is another instance of a Minkowski functional. It has the following properties:
It is subadditive:,
It is homogeneous: for all scalars ,
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.
