Minkowski functional

In mathematics, in the field of functional analysis, a Minkowski functional or gauge function is a function that recovers a notion of distance on a linear space.

If ${\displaystyle K}$ is a subset of a real or complex vector space ${\displaystyle X,}$ then the Minkowski functional or gauge of ${\displaystyle K}$ is defined to be the function ${\displaystyle p_{K}:X\to [0,\infty ],}$ valued in the extended real numbers, defined by

where the infimum of the empty set is defined to be positive infinity ${\displaystyle \,\infty \,}$ (which is not a real number so that ${\displaystyle p_{K}(x)}$ would then not be real-valued). The Minkowski function is always non-negative (meaning ${\displaystyle p_{K}\geq 0}$) and ${\displaystyle p_{K}(x)}$ is a real number if and only if ${\displaystyle \{r>0:x\in rK\}}$ 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, ${\displaystyle K}$ is usually assumed to have properties (such as being absorbing in ${\displaystyle X,}$ for instance) that will guarantee that for every ${\displaystyle x\in X,}$ this set ${\displaystyle \{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}}$ is not empty precisely because this results in ${\displaystyle p_{K}}$ being real-valued.

Moreover, ${\displaystyle K}$ is also often assumed to have more properties, such as being an absorbing disk in ${\displaystyle X,}$ since these properties guarantee that ${\displaystyle p_{K}}$ will be a (real-valued) seminorm on ${\displaystyle X.}$ In fact, every seminorm ${\displaystyle p}$ on ${\displaystyle X}$ is equal to the Minkowski functional of any subset ${\displaystyle K}$ of ${\displaystyle X}$ satisfying ${\displaystyle \{x\in X:p(x)<1\}\subseteq K\subseteq \{x\in X:p(x)\leq 1\}}$ (where all three of these sets are necessarily absorbing in ${\displaystyle X}$ and the first and last are also disks). 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). 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 geometric properties of a subset of ${\displaystyle X}$ into certain algebraic properties of a function on ${\displaystyle X.}$

Minkowski functionals have been applied to describe phase behavior of nanoconfined fluids.^[1]

Definition[]

Let ${\displaystyle K}$ be a subset of a real or complex vector space ${\displaystyle X.}$ Define the gauge of ${\displaystyle K}$ or the Minkowski functional associated with or induced by ${\displaystyle K}$ as being the function ${\displaystyle p_{K}:X\to [0,\infty ],}$ valued in the extended real numbers, defined by

where recall that the infimum of the empty set is ${\displaystyle \,\infty \,}$ (that is, ${\displaystyle \inf \varnothing =\infty }$). Here, ${\displaystyle \{r>0:x\in rK\}}$ is shorthand for ${\displaystyle \{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}.}$

For any ${\displaystyle x\in X,}$ ${\displaystyle p_{K}(x)\neq \infty }$ if and only if ${\displaystyle \{r>0:x\in rK\}}$ is not empty. The arithmetic operations on ${\displaystyle \mathbb {R} }$ can be extended to operate on ${\displaystyle \pm \infty ,}$ where ${\displaystyle {\frac {r}{\pm \infty }}:=0}$ for all non-zero real ${\displaystyle -\infty <r<\infty .}$ The products ${\displaystyle 0\cdot \infty }$ and ${\displaystyle 0\cdot -\infty }$ remain undefined.

Some conditions making a gauge real-valued

In the field of convex analysis, the map ${\displaystyle p_{K}}$ taking on the value of ${\displaystyle \,\infty \,}$ is not necessarily an issue. However, in functional analysis ${\displaystyle p_{K}}$ is almost always real-valued (that is, to never take on the value of ${\displaystyle \,\infty \,}$), which happens if and only if the set ${\displaystyle \{r>0:x\in rK\}}$ is non-empty for every ${\displaystyle x\in X.}$

In order for ${\displaystyle p_{K}}$ to be real-valued, it suffices for the origin of ${\displaystyle X}$ to belong to the algebraic interior or core of ${\displaystyle K}$ in ${\displaystyle X.}$^[2] If ${\displaystyle K}$ is absorbing in ${\displaystyle X,}$ where recall that this implies that ${\displaystyle 0\in K,}$ then the origin belongs to the algebraic interior of ${\displaystyle K}$ in ${\displaystyle X}$ and thus ${\displaystyle p_{K}}$ is real-valued. Characterizations of when ${\displaystyle p_{K}}$ is real-valued are given below.

Motivating examples[]

Example 1

Consider a normed vector space ${\displaystyle (X,\|\,\cdot \,\|),}$ with the norm ${\displaystyle \|\,\cdot \,\|}$ and let ${\displaystyle U:=\{x\in X:\|x\|\leq 1\}}$ be the unit ball in ${\displaystyle X.}$ Then for every ${\displaystyle x\in X,}$ ${\displaystyle \|x\|=p_{U}(x).}$ Thus the Minkowski functional ${\displaystyle p_{U}}$ is just the norm on ${\displaystyle X.}$

Example 2

Let ${\displaystyle X}$ be a vector space without topology with underlying scalar field ${\displaystyle \mathbb {K} .}$ Let ${\displaystyle f:X\to \mathbb {K} }$ be any linear functional on ${\displaystyle X}$ (not necessarily continuous). Fix ${\displaystyle a>0.}$ Let ${\displaystyle K}$ be the set

and let ${\displaystyle p_{K}}$ be the Minkowski functional of ${\displaystyle K.}$ Then The function ${\displaystyle p_{K}}$ has the following properties:

1. It is subadditive: ${\displaystyle p_{K}(x+y)\leq p_{K}(x)+p_{K}(y).}$
2. It is homogeneous: ${\displaystyle p_{K}(sx)=|s|p_{K}(x)}$ for all scalars ${\displaystyle s.}$
3. It is nonnegative: ${\displaystyle p_{K}\geq 0.}$

Therefore, ${\displaystyle p_{K}}$ is a seminorm on ${\displaystyle X,}$ 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, ${\displaystyle p_{K}(x)=0}$ need not imply ${\displaystyle x=0.}$ In the above example, one can take a nonzero ${\displaystyle x}$ from the kernel of ${\displaystyle f.}$ Consequently, the resulting topology need not be Hausdorff.

Common conditions guaranteeing gauges are seminorms[]

To guarantee that ${\displaystyle p_{K}(0)=0,}$ it will henceforth be assumed that ${\displaystyle 0\in K.}$

In order for ${\displaystyle p_{K}}$ to be a seminorm, it suffices for ${\displaystyle K}$ to be a disk (that is, convex and balanced) and absorbing in ${\displaystyle X,}$ which are the most common assumption placed on ${\displaystyle K.}$

More generally, if ${\displaystyle K}$ is convex and the origin belongs to the algebraic interior of ${\displaystyle K,}$ then ${\displaystyle p_{K}}$ is a nonnegative sublinear functional on ${\displaystyle X,}$ which implies in particular that it is subadditive and positive homogeneous. If ${\displaystyle K}$ is absorbing in ${\displaystyle X}$ then ${\displaystyle p_{[0,1]K}}$ is positive homogeneous, meaning that ${\displaystyle p_{[0,1]K}(sx)=sp_{[0,1]K}(x)}$ for all real ${\displaystyle s\geq 0,}$ where ${\displaystyle [0,1]K=\{tk:t\in [0,1],k\in K\}.}$^[4] If ${\displaystyle q}$ is a nonnegative real-valued function on ${\displaystyle X}$ that is positive homogeneous, then the sets ${\displaystyle U:=\{x\in X:q(x)<1\}}$ and ${\displaystyle D:=\{x\in X:q(x)\leq 1\}}$ satisfy ${\displaystyle [0,1]U=U}$ and ${\displaystyle [0,1]D=D;}$ if in addition ${\displaystyle q}$ is absolutely homogeneous then both ${\displaystyle U}$ and ${\displaystyle D}$ are balanced.^[4]

Gauges of absorbing disks[]

Arguably the most common requirements placed on a set ${\displaystyle K}$ to guarantee that ${\displaystyle p_{K}}$ is a seminorm are that ${\displaystyle K}$ be an absorbing disk in ${\displaystyle X.}$ Due to how common these assumptions are, the properties of a Minkowski functional ${\displaystyle p_{K}}$ when ${\displaystyle K}$ is an absorbing disk will now be investigated. Since all of the results mentioned above made few (if any) assumptions on ${\displaystyle K,}$ they can be applied in this special case.

Algebraic properties[]

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

• ${\displaystyle p_{K}}$ is a seminorm on ${\displaystyle X.}$
• ${\displaystyle p_{K}}$ is a norm on ${\displaystyle X}$ if and only if ${\displaystyle K}$ does not contain a non-trivial vector subspace.^[5]
• ${\displaystyle p_{sK}={\frac {1}{|s|}}p_{K}}$ for any scalar ${\displaystyle s\neq 0.}$^[5]
• If ${\displaystyle J}$ is an absorbing disk in ${\displaystyle X}$ and ${\displaystyle J\subseteq K}$ then ${\displaystyle p_{K}\leq p_{J}.}$
• If ${\displaystyle K}$ is a set satisfying ${\displaystyle \{x\in X:p(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p(x)\leq 1\}}$ then ${\displaystyle K}$ is absorbing in ${\displaystyle X}$ and ${\displaystyle p=p_{K},}$ where ${\displaystyle p_{K}}$ is the Minkowski functional associated with ${\displaystyle K;}$ that is, it is the gauge of ${\displaystyle K.}$^[6]
  • In particular, if ${\displaystyle K}$ is as above and ${\displaystyle q}$ is any seminorm on ${\displaystyle X,}$ then ${\displaystyle q=p}$ if and only if ${\displaystyle \{x\in X:q(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:q(x)\leq 1\}.}$^[6]
• If ${\displaystyle x\in X}$ satisfies ${\displaystyle p_{K}(x)<1}$ then ${\displaystyle x\in K.}$

Topological properties[]

Assume that ${\displaystyle X}$ is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let ${\displaystyle K}$ be an absorbing disk in ${\displaystyle X.}$ Then

where ${\displaystyle \operatorname {Int} _{X}K}$ is the topological interior and ${\displaystyle \operatorname {Cl} _{X}K}$ is the topological closure of ${\displaystyle K}$ in ${\displaystyle X.}$^[7] Importantly, it was not assumed that ${\displaystyle p_{K}}$ was continuous nor was it assumed that ${\displaystyle K}$ had any topological properties.

Moreover, the Minkowski functional ${\displaystyle p_{K}}$ is continuous if and only if ${\displaystyle K}$ is a neighborhood of the origin in ${\displaystyle X.}$^[7] If ${\displaystyle p_{K}}$ is continuous then^[7]

Minimal requirements on the set[]

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

Properties[]

All results in this section may be applied to the case where ${\displaystyle K}$ is an absorbing disk.

Throughout, ${\displaystyle K}$ is any subset of ${\displaystyle X.}$

Examples[]

1. If ${\displaystyle {\mathcal {L}}}$ is a non-empty collection of subsets of ${\displaystyle X}$ then ${\displaystyle p_{\cup {\mathcal {L}}}(x)=\inf \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}$ for all ${\displaystyle x\in X,}$ where ${\displaystyle \cup {\mathcal {L}}:=\bigcup _{L\in {\mathcal {L}}}L.}$
   • Thus ${\displaystyle p_{K\cup L}(x)=\min \left\{p_{K}(x),p_{L}(x)\right\}}$ for all ${\displaystyle x\in X.}$
2. If ${\displaystyle {\mathcal {L}}}$ is a non-empty collection of subsets of ${\displaystyle X}$ and ${\displaystyle I\subseteq X}$ satisfies
   $${\displaystyle \left\{x\in X:p_{L}(x)<1{\text{ for all }}L\in {\mathcal {L}}\right\}\quad \subseteq \quad I\quad \subseteq \quad \left\{x\in X:p_{L}(x)\leq 1{\text{ for all }}L\in {\mathcal {L}}\right\}}$$

   
   then ${\displaystyle p_{I}(x)=\sup \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}$ for all ${\displaystyle x\in X.}$

The following examples show that the containment ${\displaystyle (0,R]K\subseteq \bigcap _{e>0}(0,R+e)K}$ could be proper.

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

The next example shows that the containment can be proper when ${\displaystyle R=1;}$ the example may be generalized to any real ${\displaystyle R>0.}$ Assuming that ${\displaystyle [0,1]K\subseteq K,}$ the following example is representative of how it happens that ${\displaystyle x\in X}$ satisfies ${\displaystyle p_{K}(x)=1}$ but ${\displaystyle x\not \in (0,1]K.}$

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

Positive homogeneity characterizes Minkowski functionals[]

The next theorem shows that Minkowski functionals are exactly those functions ${\displaystyle f:X\to [0,\infty ]}$ that have a certain purely algebraic property that is used widely. This theorem can be extended to characterize certain classes of ${\displaystyle [-\infty ,\infty ]}$-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 ${\displaystyle f:X\to \mathbb {R} }$ (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, ${\displaystyle K}$ is not assumed to be absorbing in ${\displaystyle X}$ and instead, it is deduced that ${\displaystyle (0,1)K}$ is absorbing when ${\displaystyle p_{K}}$ is a seminorm. It is also not assumed that ${\displaystyle K}$ is balanced (which is a property that ${\displaystyle K}$ is often required to have); in its place is the weaker condition that ${\displaystyle (0,1)sK\subseteq (0,1)K}$ for all scalars ${\displaystyle s}$ satisfying ${\displaystyle |s|=1.}$ The common requirement that ${\displaystyle K}$ be convex is also weakened to only requiring that ${\displaystyle (0,1)K}$ be convex.

Positive sublinear functions and Minkowski functionals[]

It may be shown that a real-valued subadditive function ${\displaystyle f:X\to \mathbb {R} }$ on an arbitrary topological vector space ${\displaystyle X}$ is continuous at the origin if and only if it is uniformly continuous, where if in addition ${\displaystyle f}$ is nonnegative, then ${\displaystyle f}$ is continuous if and only if ${\displaystyle V:=\{x\in X:f(x)<1\}}$ is an open neighborhood in ${\displaystyle X.}$^[8] If ${\displaystyle f:X\to \mathbb {R} }$ is subadditive and satisfies ${\displaystyle f(0)=0,}$ then ${\displaystyle f}$ is continuous if and only if its absolute value ${\displaystyle |f|:X\to [0,\infty )}$ is continuous.

A positive sublinear function is a nonnegative homogeneous function ${\displaystyle f:X\to [0,\infty )}$ that satisfies the triangle inequality. It follows immediately from the results below that for such a function ${\displaystyle f,}$ if ${\displaystyle V:=\{x\in X:f(x)<1\}}$ then ${\displaystyle f=p_{V}.}$ Given ${\displaystyle K\subseteq X,}$ the Minkowski functional ${\displaystyle p_{K}}$ is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if ${\displaystyle (0,\infty )K=X}$ and ${\displaystyle (0,1)K}$ is convex.

Correspondence between open convex sets and positive continuous sublinear functions

See also[]

• Asymmetric norm – Generalization of the concept of a norm
• Cauchy's functional equation – Functional equation
• Finest locally convex topology
• Finsler manifold
• Hadwiger's theorem
• Hugo Hadwiger – Swiss mathematician
• Locally convex topological vector space – A vector space with a topology defined by convex open sets
• Morphological image processing
• Norm (mathematics) – Length in a vector space
• Seminorm
• Topological vector space – Vector space with a notion of nearness

Notes[]

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