Minkowski Functional

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 p_K : X \to , \infty valued in the extended real numbers, defined by
$p_K(x) := \inf \ \quad \text x \in X,$
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 Minkowski function is always non-negative (meaning $p_K \geq 0$) and $p_K(x)$ is a real number if and only if $\$ is not empty.
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.

In functional analysis, $K$ is usually assumed to have properties (such as being absorbing in $X,$ for instance) that will guarantee that for every $x \in X,$ this set $\$ is not empty precisely because this results in $p_K$ being real-valued.

Moreover, $K$ is also often assumed to have more properties, such as being an absorbing disk in $X,$ since these properties 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 of any subset $K$ of $X$ satisfying $\ \subseteq K \subseteq \$ (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 $X.$

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 p_K : X \to , \infty valued in the extended real numbers, defined by
$p_K(x) := \inf \,$
where recall that the infimum of the empty set is $\,\infty\,$ (that is, $\inf \varnothing = \infty$). Here, $\$ is shorthand for $\.$

For any $x \in X,$ $p_K(x) \neq \infty$ if and only if $\$ is not empty.
The arithmetic operations on $\R$ can be extended to operate on $\pm \infty,$ where $\frac := 0$ for all non-zero real $- \infty < r < \infty.$
The products $0 \cdot \infty$ and $0 \cdot - \infty$ 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 $(X, \, \,\cdot\,\, ),$ with the norm $\, \,\cdot\,\,$ and let $U := \$ be the unit ball in $X.$ Then for every $x \in X,$ $\, x\, = p_U(x).$ 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 $\mathbb.$
Let $f : X \to \mathbb$ be any linear functional on $X$ (not necessarily continuous).
Fix $a > 0.$
Let $K$ be the set
$K := \$
and let $p_K$ be the Minkowski functional of $K.$
Then
$p_K(x) = \frac , f(x), \quad \text x \in X.$
The function $p_K$ has the following properties:

#It is : $p_K(x + y) \leq p_K(x) + p_K(y).$
#It is : $p_K(s x) = , s, p_K(x)$ for all scalars $s.$
#It is : $p_K \geq 0.$

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 $p_$ is positive homogeneous, meaning that $p_(s x) = s p_(x)$ for all real $s \geq 0,$ where $$, 1K = \.
If $q$ is a nonnegative real-valued function on $X$ that is positive homogeneous, then the sets $U := \$ and $D := \$ satisfy $$, 1U = U and $$, 1D = D;
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 $p_K(x) = p_K(y) = r.$
Then for all $e > 0,$ $x, y \in K_e := (r, e) K.$
Since $K$ is convex and $r + e \neq 0,$ $K_e$ is also convex.
Therefore, $\frac x + \frac y \in K_e.$
By definition of the Minkowski functional $p_K,$
$p_K\left(\frac x + \frac y\right) \leq r + e = \frac p_K(x) + \frac p_K(y) + e.$

But the left hand side is $\frac p_K(x + y),$ so that
$p_K(x + y) \leq p_K(x) + p_K(y) + 2 e.$

Since $e > 0$ was arbitrary, it follows that $p_K(x + y) \leq p_K(x) + p_K(y),$ which is the desired inequality.
The general case $p_K(x) > p_K(y)$ 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
$\lambda x \in r K \quad \mbox \quad x \in \frac K.$

Therefore
$p_K (\lambda x) = \inf \left\ = \inf \left\ = \inf \left\ = , \lambda, p_K(x).$

Algebraic properties

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

• $p_K$ is a seminorm on $X.$


• $p_K$ is a norm on $X$ if and only if $K$ does not contain a non-trivial vector subspace.


• $p_ = \frac p_K$ for any scalar $s \neq 0.$


• If $J$ is an absorbing disk in $X$ and $J \subseteq K$ then $p_K \leq p_J.$


• If $K$ is a set satisfying $\ \; \subseteq \; K \; \subseteq \; \$ then $K$ is absorbing in $X$ and $p = p_K,$ where $p_K$ is the Minkowski functional associated with $K;$ that is, it is the gauge of $K.$
  * In particular, if $K$ is as above and $q$ is any seminorm on $X,$ then $q = p$ if and only if $\ \; \subseteq \; K \; \subseteq \; \.$


• If $x \in X$ satisfies $p_K(x) < 1$ then $x \in K.$

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
$\operatorname_X K \; \subseteq \; \ \; \subseteq \; K \; \subseteq \; \ \; \subseteq \; \operatorname_X K,$
where $\operatorname_X K$ is the topological interior and $\operatorname_X K$ 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
$\operatorname_X K = \ \quad \text \quad \operatorname_X K = \.$

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 $A \subseteq X$ that satisfies $(0, \infty) A = X$ is necessarily absorbing in $X$ is straightforward and can be found in the article on absorbing sets.

For any real $t > 0,$
$\ = \left\ = t \$
so that taking the infimum of both sides shows that
$p_K(tx) = \inf \ = t \inf \ = t p_K(x).$
This proves that Minkowski functionals are strictly positive homogeneous. For $0 \cdot p_K(x)$ to be well-defined, it is necessary and sufficient that $p_K(x) \neq \infty;$ thus $p_K(tx) = t p_K(x)$ 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 $p_K(s x) = p_K(x)$ for all $x \in X$ and all scalars $s$ satisfying $, s, = 1.$
Every scalar $s$ is of the form $r e^$ for some real $t$ where $r := , s, \geq 0$ and $e^$ 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.
$\blacksquare$

Examples

1. If $\mathcal$ is a non-empty collection of subsets of $X$ then $p_(x) = \inf \left\$ for all $x \in X,$ where $\cup \mathcal := \bigcup_ L.$
   * Thus $p_(x) = \min \left\$ for all $x \in X.$
   


2. If $\mathcal$ is a non-empty collection of subsets of $X$ and $I \subseteq X$ satisfies
   $\left\ \quad \subseteq \quad I \quad \subseteq \quad \left\$
   then $p_I(x) = \sup \left\$ for all $x \in X.$
   

The following examples show that the containment $(0, R] K \subseteq \bigcap_ (0, R + e) K$ could be proper.

Example: If $R = 0$ and $K = X$ then $(0, R] K = (0, 0] X = \varnothing X = \varnothing$ but $\bigcap_ (0, e) K = \bigcap_ X = X,$ which shows that its possible for $(0, R] K$ to be a proper subset of $\bigcap_ (0, R + e) K$ when $R = 0.$ $\blacksquare$

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 $$, 1K \subseteq K, the following example is representative of how it happens that $x \in X$ satisfies $p_K(x) = 1$ but $x \not\in (0, 1] K.$

Example: Let $x \in X$ be non-zero and let $K = [0, 1) x$ so that $$, 1K = K and $x \not\in K.$
From $x \not\in (0, 1) K = K$ it follows that $p_K(x) \geq 1.$
That $p_K(x) \leq 1$ follows from observing that for every $e > 0,$ $(0, 1 + e) K = [0, 1 + e)([0, 1) x) = [0, 1 + e) x,$ which contains $x.$
Thus $p_K(x) = 1$ and $x \in \bigcap_ (0, 1 + e) K.$
However, $(0, 1] K = (0, 1]([0, 1) x) = [0, 1) x = K$ so that $x \not\in (0, 1] K,$ as desired.
$\blacksquare$

Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are those functions f : X \to , \infty/math> that have a certain purely algebraic property that is used widely.
This theorem can be extended to characterize certain classes of \infty, \infty/math>-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 $f : X \to \R$ (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

If $f(t x) \leq t f(x)$ holds for all $x \in X$ and real $t > 0$ then $t f(x) = t f\left(\tfrac(t x)\right) \leq t \tfrac f(t x) = f(t x) \leq t f(x)$ so that $t f(x) = f(t x).$

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 f : X \to , \infty/math> is a function such that $f(t x) = t f(x)$ for all $x \in X$ and all real $t > 0$ and let $K := \.$

For all real $t > 0,$ $f(0) = f(t 0) = t f(0)$ so by taking $t = 2$ for instance, it follows that either $f(0) = 0 \text f(0) = \infty.$
Let $x \in X.$
It remains to show that $f(x) = p_K(x).$

It will now be shown that if $f(x) = 0$ or $f(x) = \infty$ then $f(x) = p_K(x),$ so that in particular, it will follow that $f(0) = p_K(0).$
So suppose that $f(x) = 0$ or $f(x) = \infty;$ in either case $f(t x) = t f(x) = f(x)$ 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 $f(t x) = 0 \leq 1$), which implies that $p_K(x) = 0,$ as desired.
Similarly, if $f(x) = \infty$ then $t x \not\in K$ for all real $t > 0,$ which implies that $p_K(x) = \infty,$ 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 $p_K(t x) = t p_K(x)$ for all real $t > 0.$
Since $0 < \tfrac < \infty,$ $p_K(x)= R = f(x)$ if and only if $p_K\left(\tfrac x\right) = 1 = f\left(\tfrac x\right)$ so assume without loss of generality that $R = 1$ and it remains to show that $p_K\left(\tfrac x\right) = 1.$
Since $f(x) = 1,$ $x \in K \subseteq (0, 1] K,$ which implies that $p_K(x) \leq 1$ (so in particular, $p_K(x) \neq \infty$ is guaranteed).
It remains to show that $p_K(x) \geq 1,$ which recall happens if and only if $x \not\in (0, 1) K.$
So assume for the sake of contradiction that $x \in (0, 1) K$ and let $0 < r < 1$ and $k \in K$ be such that $x = r k,$ where note that $k \in K$ implies that $f(k) \leq 1.$
Then $1 = f(x) = f(r k) = r f(k) \leq r < 1.$ $\blacksquare$

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 $(0, 1) s K \subseteq (0, 1) K$ 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 $f : X \to \R$ 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 $V := \$ is an open neighborhood in $X.$
If $f : X \to \R$ is subadditive and satisfies $f(0) = 0,$ then $f$ is continuous if and only if its absolute value , f, : X \to sublinear_function_In_linear_algebra,_a_sublinear_function_(or_functional_as_is_more_often_used_in_functional_analysis),_also_called_a_quasi-seminorm_or_a_Banach_functional,_on_a_vector_space_X_is_a_real-valued_function_with_only_some_of_the_properties_of_a_seminorm.__...