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 K is a subset of a real or complex vector space X, then the or of K 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 ,\infty, (which is a real number so that p_K(x) would then be real-valued). The set K is often assumed/picked to have properties, such as being an absorbing disk in X, that guarantee that p_K will be a real-valued seminorm on X. In fact, every seminorm p on X is equal to the Minkowski functional (that is, p = p_K) of any subset K of X satisfying (where all three of these sets are necessarily absorbing in X 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 properties of a subset of X into certain properties of a function on X. The Minkowski function is always non-negative (meaning p_K \geq 0). 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, p_K might not be real-valued since for any given x \in X, the value p_K(x) is a real number if and only if is not empty. Consequently, K is usually assumed to have properties (such as being absorbing in X, for instance) that will guarantee that p_K is real-valued.

Definition

Let K be a subset of a real or complex vector space X. Define the of K or the associated with or induced by K as being the function valued in the extended real numbers, defined by (recall that the infimum of the empty set is ,\infty, that is, ). Here, is shorthand for For any x \in X, if and only if is not empty. The arithmetic operations on \R can be extended to operate on \pm \infty, 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 p_K taking on the value of ,\infty, is not necessarily an issue. However, in functional analysis p_K is almost always real-valued (that is, to never take on the value of ,\infty,), which happens if and only if the set is non-empty for every x \in X. In order for p_K to be real-valued, it suffices for the origin of X to belong to the or of K in X. If K is absorbing in X, where recall that this implies that 0 \in K, then the origin belongs to the algebraic interior of K in X and thus p_K is real-valued. Characterizations of when p_K 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 X. Then for every x \in X, Thus the Minkowski functional p_U is just the norm on X.

Example 2

Let X be a vector space without topology with underlying scalar field \mathbb{K}. Let be any linear functional on X (not necessarily continuous). Fix a > 0. Let K be the set and let p_K be the Minkowski functional of K. Then The function p_K has the following properties: Therefore, p_K is a seminorm on X, 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, p_K(x) = 0 need not imply x = 0. In the above example, one can take a nonzero x from the kernel of f. Consequently, the resulting topology need not be Hausdorff.

Common conditions guaranteeing gauges are seminorms

To guarantee that p_K(0) = 0, it will henceforth be assumed that 0 \in K. In order for p_K to be a seminorm, it suffices for K to be a disk (that is, convex and balanced) and absorbing in X, which are the most common assumption placed on K. More generally, if K is convex and the origin belongs to the algebraic interior of K, then p_K is a nonnegative sublinear functional on X, which implies in particular that it is subadditive and positive homogeneous. If K is absorbing in X then is positive homogeneous, meaning that for all real s \geq 0, where If q is a nonnegative real-valued function on X that is positive homogeneous, then the sets and satisfy and if in addition q is absolutely homogeneous then both U and D are balanced.

Gauges of absorbing disks

Arguably the most common requirements placed on a set K to guarantee that p_K is a seminorm are that K be an absorbing disk in X. Due to how common these assumptions are, the properties of a Minkowski functional p_K when K is an absorbing disk will now be investigated. Since all of the results mentioned above made few (if any) assumptions on K, they can be applied in this special case. Convexity and subadditivity A simple geometric argument that shows convexity of K implies subadditivity is as follows. Suppose for the moment that Then for all e > 0, Since K is convex and K_e is also convex. Therefore, By definition of the Minkowski functional p_K, But the left hand side is so that Since e > 0 was arbitrary, it follows that which is the desired inequality. The general case is obtained after the obvious modification. Convexity of K, together with the initial assumption that the set is nonempty, implies that K is absorbing. Balancedness and absolute homogeneity Notice that K being balanced implies that Therefore

Algebraic properties

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

Topological properties

Assume that X is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let K be an absorbing disk in X. Then where is the topological interior and is the topological closure of K in X. Importantly, it was assumed that p_K was continuous nor was it assumed that K had any topological properties. Moreover, the Minkowski functional p_K is continuous if and only if K is a neighborhood of the origin in X. If p_K is continuous then

Minimal requirements on the set

This section will investigate the most general case of the gauge of subset K of X. The more common special case where K is assumed to be an absorbing disk in X was discussed above.

Properties

All results in this section may be applied to the case where K is an absorbing disk. Throughout, K is any subset of X. 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 X is straightforward and can be found in the article on absorbing sets. For any real t > 0, 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 x \in X and all real t \geq 0 if and only if p_K is real-valued. The hypothesis of statement (7) allows us to conclude that for all x \in X and all scalars s satisfying |s| = 1. Every scalar s is of the form r e^{i t} for some real t where and e^{i t} is real if and only if s is real. The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of p_K, and from the positive homogeneity of p_K when p_K is real-valued.

Examples

then for all x \in X. The following examples show that the containment could be proper. Example: If R = 0 and K = X then but which shows that its possible for (0, R] K to be a proper subset of when R = 0. The next example shows that the containment can be proper when R = 1; the example may be generalized to any real R > 0. Assuming that the following example is representative of how it happens that x \in X satisfies p_K(x) = 1 but Example: Let x \in X be non-zero and let so that and From it follows that That follows from observing that for every e > 0, which contains x. Thus p_K(x) = 1 and However, so that as desired.

Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are those functions that have a certain purely algebraic property that is commonly encountered. If holds for all x \in X and real t > 0 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 x \in X and all real t > 0 and let For all real t > 0, so by taking t = 2 for instance, it follows that either f(0) = 0 or Let x \in X. It remains to show that It will now be shown that if f(x) = 0 or then so that in particular, it will follow that So suppose that f(x) = 0 or in either case for all real t > 0. Now if f(x) = 0 then this implies that that t x \in K for all real t > 0 (since ), which implies that p_K(x) = 0, as desired. Similarly, if then for all real t > 0, which implies that as desired. Thus, it will henceforth be assumed that R := f(x) a positive real number and that x \neq 0 (importantly, however, the possibility that p_K(x) is 0 or ,\infty, has not yet been ruled out). Recall that just like f, the function p_K satisfies for all real t > 0. Since if and only if so assume without loss of generality that R = 1 and it remains to show that Since f(x) = 1, 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 0 < r < 1 and k \in K be such that x = r k, where note that k \in K 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

Characterizing Minkowski functionals that are seminorms

In this next theorem, which follows immediately from the statements above, K is assumed to be absorbing in X and instead, it is deduced that (0, 1) K is absorbing when p_K is a seminorm. It is also not assumed that K is balanced (which is a property that K is often required to have); in its place is the weaker condition that for all scalars s satisfying |s| = 1. The common requirement that K be convex is also weakened to only requiring that (0, 1) K be convex.

Positive sublinear functions and Minkowski functionals

It may be shown that a real-valued subadditive function on an arbitrary topological vector space X is continuous at the origin if and only if it is uniformly continuous, where if in addition f is nonnegative, then f is continuous if and only if is an open neighborhood in X. If is subadditive and satisfies f(0) = 0, then f is continuous if and only if its absolute value is continuous. A is a nonnegative homogeneous function that satisfies the triangle inequality. It follows immediately from the results below that for such a function f, if then f = p_V. Given the Minkowski functional p_K is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if and (0, 1) K is convex.

Correspondence between open convex sets and positive continuous sublinear functions

Let be an open convex subset of X. If 0 \in V then let z := 0 and otherwise let z \in V be arbitrary. Let be the Minkowski functional of K := V - z where this convex open neighborhood of the origin satisfies Then p is a continuous sublinear function on X since V - z is convex, absorbing, and open (however, p 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.