Minkowski functional explained

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 $p\_K\; :\; X\; \backslash to\; [0,\; \backslash infty],$ valued in the extended real numbers, defined by$$p\_K(x)\; :=\; \backslash inf\; \backslash \; \backslash quad\; \backslash text\; x\; \backslash in\; X,$$where the infimum of the empty set is defined to be positive infinity $\backslash ,\backslash infty\backslash ,$ (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

$$\backslash \; \backslash subseteq\; K\; \backslash subseteq\; \backslash$$

(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 $p\_K\; \backslash geq\; 0$). 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 $x\; \backslash in\; X,$ the value $p\_K(x)$ is a real number if and only if $\backslash$ 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 $p\_K\; :\; X\; \backslash to\; [0,\; \backslash infty],$ valued in the extended real numbers, defined by

$$p\_K(x)\; :=\; \backslash inf\; \backslash ,$$

(recall that the infimum of the empty set is $\backslash ,\backslash infty$, that is, $\backslash inf\; \backslash varnothing\; =\; \backslash infty$). Here, $\backslash$ is shorthand for $\backslash .$

For any $x\; \backslash in\; X,$ $p\_K(x)\; \backslash neq\; \backslash infty$ if and only if $\backslash$ is not empty. The arithmetic operations on $\backslash R$ can be extended to operate on $\backslash pm\; \backslash infty,$ where $\backslash frac\; :=\; 0$ for all non-zero real The products $0\; \backslash cdot\; \backslash infty$ and $0\; \backslash cdot\; -\; \backslash infty$ remain undefined.

Some conditions making a gauge real-valued

In the field of convex analysis, the map $p\_K$ taking on the value of $\backslash ,\backslash infty\backslash ,$ 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 $\backslash ,\backslash infty\backslash ,$), which happens if and only if the set $\backslash$ is non-empty for every $x\; \backslash in\; X.$

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 $0\; \backslash in\; K,$ 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 $(X,\; \backslash |\backslash ,\backslash cdot\backslash ,\backslash |),$ with the norm $\backslash |\backslash ,\backslash cdot\backslash ,\backslash |$ and let $U\; :=\; \backslash$ be the unit ball in $X.$ Then for every $x\; \backslash in\; X,$ $\backslash |x\backslash |\; =\; p\_U(x).$ 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 $\backslash mathbb.$Let $f\; :\; X\; \backslash to\; \backslash mathbb$ be any linear functional on $X$ (not necessarily continuous). Fix $a\; >\; 0.$ Let $K$ be the set $$K\; :=\; \backslash$$and let $p\_K$ be the Minkowski functional of $K.$ Then$$p\_K(x)\; =\; \backslash frac\; |f(x)|\; \backslash quad\; \backslash text\; x\; \backslash in\; X.$$The function $p\_K$ has the following properties:

1. It is : $p\_K(x\; +\; y)\; \backslash leq\; p\_K(x)\; +\; p\_K(y).$
2. It is : $p\_K(s\; x)\; =\; |s|\; p\_K(x)$ for all scalars $s.$
3. It is : $p\_K\; \backslash geq\; 0.$

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 $0\; \backslash in\; K.$

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

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

Since $e\; >\; 0$ was arbitrary, it follows that $p\_K(x\; +\; y)\; \backslash leq\; p\_K(x)\; +\; p\_K(y),$ 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 $\backslash$ is nonempty, implies that $K$ is absorbing.

Balancedness and absolute homogeneity

Notice that $K$ being balanced implies that$$\backslash lambda\; x\; \backslash in\; r\; K\; \backslash quad\; \backslash mbox\; \backslash quad\; x\; \backslash in\; \backslash frac$$

K.

Therefore$$p\_K\; (\backslash lambda\; x)\; =\; \backslash inf\; \backslash left\backslash \; =\; \backslash inf\; \backslash left\backslash =\; \backslash inf\; \backslash left\backslash$$

\frac > 0 : x \in \frac K \right\

= |\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.
• $p\_\; =\; \backslash frac$

p_K for any scalar $s\; \backslash neq\; 0.$

• If $J$ is an absorbing disk in $X$ and $J\; \backslash subseteq\; K$ then $p\_K\; \backslash leq\; p\_J.$
• If $K$ is a set satisfying $\backslash \; \backslash ;\; \backslash subseteq\; \backslash ;\; K\; \backslash ;\; \backslash subseteq\; \backslash ;\; \backslash$ 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 $\backslash \; \backslash ;\; \backslash subseteq\; \backslash ;\; K\; \backslash ;\; \backslash subseteq\; \backslash ;\; \backslash .$
• If $x\; \backslash in\; X$ satisfies then $x\; \backslash in\; K.$

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

$$\backslash operatorname\_X\; K\; \backslash ;\; \backslash subseteq\; \backslash ;\; \backslash \; \backslash ;\; \backslash subseteq\; \backslash ;\; K\; \backslash ;\; \backslash subseteq\; \backslash ;\; \backslash \; \backslash ;\; \backslash subseteq\; \backslash ;\; \backslash operatorname\_X\; K,$$

where $\backslash operatorname\_X\; K$ is the topological interior and $\backslash operatorname\_X\; K$ 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$$\backslash operatorname\_X\; K\; =\; \backslash \; \backslash quad\; \backslash text\; \backslash quad\; \backslash operatorname\_X\; K\; =\; \backslash .$$

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

For any real $t\; >\; 0,$

$$\backslash \; =\; \backslash \; =\; t\; \backslash$$

so that taking the infimum of both sides shows that

$$p\_K(tx)\; =\; \backslash inf\; \backslash \; =\; t\; \backslash inf\; \backslash \; =\; t\; p\_K(x).$$

This proves that Minkowski functionals are strictly positive homogeneous. For $0\; \backslash cdot\; p\_K(x)$ to be well-defined, it is necessary and sufficient that $p\_K(x)\; \backslash neq\; \backslash infty;$ thus $p\_K(tx)\; =\; t\; p\_K(x)$ for all $x\; \backslash in\; X$ and all real $t\; \backslash geq\; 0$ 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 $x\; \backslash in\; X$ and all scalars $s$ satisfying $|s|\; =\; 1.$ Every scalar $s$ is of the form $r\; e^$ for some real $t$ where $r\; :=\; |s|\; \backslash geq\; 0$ 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. $\backslash blacksquare$

Examples

1. If $\backslash mathcal$ is a non-empty collection of subsets of $X$ then $p\_(x)\; =\; \backslash inf\; \backslash left\backslash$ for all $x\; \backslash in\; X,$ where $\backslash cup\; \backslash mathcal\; ~\backslash stackrel~\; L.$
   • Thus $p\_(x)\; =\; \backslash min\; \backslash left\backslash$ for all $x\; \backslash in\; X.$
2. If $\backslash mathcal$ is a non-empty collection of subsets of $X$ and $I\; \backslash subseteq\; X$ satisfies

$$\backslash left\backslash \; \backslash quad\; \backslash subseteq\; \backslash quad\; I\; \backslash quad\; \backslash subseteq\; \backslash quad\; \backslash left\backslash$$ then $p\_I(x)\; =\; \backslash sup\; \backslash left\backslash$ for all $x\; \backslash in\; X.$

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

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

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

Example: Let $x\; \backslash in\; X$ be non-zero and let and $x\; \backslash not\backslash in\; K.$ From $x\; \backslash not\backslash in\; (0,\; 1)\; K\; =\; K$ it follows that $p\_K(x)\; \backslash geq\; 1.$ That $p\_K(x)\; \backslash leq\; 1$ follows from observing that for every $e\; >\; 0,$ as desired. $\backslash blacksquare$

Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are those functions $f\; :\; X\; \backslash to\; [0,\; \backslash infty]$ that have a certain purely algebraic property that is commonly encountered.

If $f(t\; x)\; \backslash leq\; t\; f(x)$ holds for all $x\; \backslash in\; X$ and real $t\; >\; 0$ then $t\; f(x)\; =\; t\; f\backslash left(\backslash tfrac(t\; x)\backslash right)\; \backslash leq\; t\; \backslash tfrac\; f(t\; x)\; =\; f(t\; x)\; \backslash leq\; t\; f(x)$ 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 $f\; :\; X\; \backslash to\; [0,\; \backslash infty]$ is a function such that $f(t\; x)\; =\; t\; f(x)$ for all $x\; \backslash in\; X$ and all real $t\; >\; 0$ and let $K\; :=\; \backslash .$

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

It will now be shown that if $f(x)\; =\; 0$ or $f(x)\; =\; \backslash infty$ 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 $f(x)\; =\; \backslash infty;$ 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 $t\; x\; \backslash in\; K$ for all real $t\; >\; 0$ (since $f(t\; x)\; =\; 0\; \backslash leq\; 1$), which implies that $p\_K(x)\; =\; 0,$ as desired. Similarly, if $f(x)\; =\; \backslash infty$ then $t\; x\; \backslash not\backslash in\; K$ for all real $t\; >\; 0,$ which implies that $p\_K(x)\; =\; \backslash infty,$ as desired. Thus, it will henceforth be assumed that $R\; :=\; f(x)$ a positive real number and that $x\; \backslash neq\; 0$ (importantly, however, the possibility that $p\_K(x)$ is $0$ or $\backslash ,\backslash infty\backslash ,$ 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 $p\_K(x)=\; R\; =\; f(x)$ if and only if $p\_K\backslash left(\backslash tfrac\; x\backslash right)\; =\; 1\; =\; f\backslash left(\backslash tfrac\; x\backslash right)$ so assume without loss of generality that $R\; =\; 1$ and it remains to show that $p\_K\backslash left(\backslash tfrac\; x\backslash right)\; =\; 1.$ Since $f(x)\; =\; 1,$ $x\; \backslash in\; K\; \backslash subseteq\; (0,\; 1]\; K,$ which implies that $p\_K(x)\; \backslash leq\; 1$ (so in particular, $p\_K(x)\; \backslash neq\; \backslash infty$ is guaranteed). It remains to show that $p\_K(x)\; \backslash geq\; 1,$ which recall happens if and only if $x\; \backslash not\backslash in\; (0,\; 1)\; K.$ So assume for the sake of contradiction that $x\; \backslash in\; (0,\; 1)\; K$ and let and $k\; \backslash in\; K$ be such that $x\; =\; r\; k,$ where note that $k\; \backslash in\; K$ implies that $f(k)\; \backslash leq\; 1.$ Then $\backslash blacksquare$

This theorem can be extended to characterize certain classes of $[-\; \backslash infty,\; \backslash infty]$-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 $f\; :\; X\; \backslash to\; \backslash R$ (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

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 $(0,\; 1)\; s\; K\; \backslash subseteq\; (0,\; 1)\; K$ 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 $f\; :\; X\; \backslash to\; \backslash R$ 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 $V\; :=\; \backslash$ is an open neighborhood in $X.$ If $f\; :\; X\; \backslash to\; \backslash R$ is subadditive and satisfies $f(0)\; =\; 0,$ then $f$ is continuous if and only if its absolute value

Let $V\; \backslash neq\; \backslash varnothing$ be an open convex subset of $X.$ If $0\; \backslash in\; V$ then let $z\; :=\; 0$ and otherwise let $z\; \backslash in\; V$ be arbitrary. Let

• • • Book: Schaefer, H. H.. Topological Vector Spaces. Springer New York Imprint Springer. New York, NY. 1999. 978-1-4612-7155-0. 840278135.

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). .