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 ${\textstyle K}$ is a subset of a real or complex vector space ${\textstyle X,}$ then the Minkowski functional or gauge of ${\textstyle K}$ is defined to be the function ${\textstyle p_{K}:X\to [0,\infty ],}$ valued in the extended real numbers, defined by $${\displaystyle p_{K}(x)=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\},\quad x\in X,}$$ where the infimum of the empty set is defined to be positive infinity.

The set ${\textstyle K}$ is often assumed to have properties, such as being an absorbing disk in ${\textstyle X}$, that guarantee that ${\textstyle p_{K}}$ will be a seminorm on ${\textstyle X.}$ In fact, every seminorm ${\textstyle p}$ on ${\textstyle X}$ is equal to the Minkowski functional (that is, ${\textstyle p=p_{K}}$) of any subset ${\textstyle K}$ of ${\textstyle X}$ 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 ${\textstyle X}$ 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 ${\textstyle X}$ into certain algebraic properties of a function on ${\textstyle X.}$

The Minkowski function is always non-negative (meaning ${\textstyle p_{K}\geq 0}$). 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, ${\textstyle p_{K}}$ might not be real-valued since for any given ${\textstyle x\in X,}$ the value ${\textstyle p_{K}(x)}$ is a real number if and only if ${\textstyle \{r>0:x\in rK\}}$ is not empty. Consequently, ${\textstyle K}$ is usually assumed to have properties (such as being absorbing in ${\textstyle X,}$ for instance) that will guarantee that ${\textstyle p_{K}}$ is real-valued.

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

$${\displaystyle p_{K}(x):=\inf\{r>0:x\in rK\},}$$

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

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

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

In order for ${\textstyle p_{K}}$ to be real-valued, it suffices for the origin of ${\textstyle X}$ to belong to the algebraic interior or core of ${\textstyle K}$ in ${\textstyle X.}$¹ If ${\textstyle K}$ is absorbing in ${\textstyle X,}$ where recall that this implies that ${\textstyle 0\in K,}$ then the origin belongs to the algebraic interior of ${\textstyle K}$ in ${\textstyle X}$ and thus ${\textstyle p_{K}}$ is real-valued. Characterizations of when ${\textstyle p_{K}}$ is real-valued are given below.

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

Let ${\textstyle X}$ be a vector space without topology with underlying scalar field ${\textstyle \mathbb {K} .}$ Let ${\textstyle f:X\to \mathbb {K} }$ be any linear functional on ${\textstyle X}$ (not necessarily continuous). Fix ${\textstyle a>0.}$ Let ${\textstyle K}$ be the set $${\displaystyle K:=\{x\in X:|f(x)|\leq a\}}$$ and let ${\textstyle p_{K}}$ be the Minkowski functional of ${\textstyle K.}$ Then $${\displaystyle p_{K}(x)={\frac {1}{a}}|f(x)|\quad {\text{ for all }}x\in X.}$$ The function ${\textstyle p_{K}}$ has the following properties:

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

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

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

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

More generally, if ${\textstyle K}$ is convex and the origin belongs to the algebraic interior of ${\textstyle K,}$ then ${\textstyle p_{K}}$ is a nonnegative sublinear functional on ${\textstyle X,}$ which implies in particular that it is subadditive and positive homogeneous. If ${\textstyle K}$ is absorbing in ${\textstyle X}$ then ${\textstyle p_{[0,1]K}}$ is positive homogeneous, meaning that ${\textstyle p_{[0,1]K}(sx)=sp_{[0,1]K}(x)}$ for all real ${\textstyle s\geq 0,}$ where ${\textstyle [0,1]K=\{tk:t\in [0,1],k\in K\}.}$³ If ${\textstyle q}$ is a nonnegative real-valued function on ${\textstyle X}$ that is positive homogeneous, then the sets ${\textstyle U:=\{x\in X:q(x)<1\}}$ and ${\textstyle D:=\{x\in X:q(x)\leq 1\}}$ satisfy ${\textstyle [0,1]U=U}$ and ${\textstyle [0,1]D=D;}$ if in addition ${\textstyle q}$ is absolutely homogeneous then both ${\textstyle U}$ and ${\textstyle D}$ are balanced.³

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

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

• ${\textstyle p_{K}}$ is a seminorm on ${\textstyle X.}$
• ${\textstyle p_{K}}$ is a norm on ${\textstyle X}$ if and only if ${\textstyle K}$ does not contain a non-trivial vector subspace.⁴
• ${\textstyle p_{sK}={\frac {1}{|s|}}p_{K}}$ for any scalar ${\textstyle s\neq 0.}$⁴
• If ${\textstyle J}$ is an absorbing disk in ${\textstyle X}$ and ${\textstyle J\subseteq K}$ then ${\textstyle p_{K}\leq p_{J}.}$
• If ${\textstyle K}$ is a set satisfying ${\textstyle \{x\in X:p(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p(x)\leq 1\}}$ then ${\textstyle K}$ is absorbing in ${\textstyle X}$ and ${\textstyle p=p_{K},}$ where ${\textstyle p_{K}}$ is the Minkowski functional associated with ${\textstyle K;}$ that is, it is the gauge of ${\textstyle K.}$⁵
• In particular, if ${\textstyle K}$ is as above and ${\textstyle q}$ is any seminorm on ${\textstyle X,}$ then ${\textstyle q=p}$ if and only if ${\textstyle \{x\in X:q(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:q(x)\leq 1\}.}$⁵
• If ${\textstyle x\in X}$ satisfies ${\textstyle p_{K}(x)<1}$ then ${\textstyle x\in K.}$

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

$${\displaystyle \operatorname {Int} _{X}K\;\subseteq \;\{x\in X:p_{K}(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p_{K}(x)\leq 1\}\;\subseteq \;\operatorname {Cl} _{X}K,}$$

where ${\textstyle \operatorname {Int} _{X}K}$ is the topological interior and ${\textstyle \operatorname {Cl} _{X}K}$ is the topological closure of ${\textstyle K}$ in ${\textstyle X.}$⁶ Importantly, it was not assumed that ${\textstyle p_{K}}$ was continuous nor was it assumed that ${\textstyle K}$ had any topological properties.

Moreover, the Minkowski functional ${\textstyle p_{K}}$ is continuous if and only if ${\textstyle K}$ is a neighborhood of the origin in ${\textstyle X.}$⁶ If ${\textstyle p_{K}}$ is continuous then⁶ $${\displaystyle \operatorname {Int} _{X}K=\{x\in X:p_{K}(x)<1\}\quad {\text{ and }}\quad \operatorname {Cl} _{X}K=\{x\in X:p_{K}(x)\leq 1\}.}$$

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

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

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

1. If ${\textstyle {\mathcal {L}}}$ is a non-empty collection of subsets of ${\textstyle X}$ then ${\textstyle p_{\cup {\mathcal {L}}}(x)=\inf \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}$ for all ${\textstyle x\in X,}$ where ${\textstyle \cup {\mathcal {L}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \bigcup \limits _{L\in {\mathcal {L}}}}L.}$
   • Thus ${\textstyle p_{K\cup L}(x)=\min \left\{p_{K}(x),p_{L}(x)\right\}}$ for all ${\textstyle x\in X.}$
2. If ${\textstyle {\mathcal {L}}}$ is a non-empty collection of subsets of ${\textstyle X}$ and ${\textstyle I\subseteq X}$ 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 ${\textstyle p_{I}(x)=\sup \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}$ for all ${\textstyle x\in X.}$

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

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

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

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

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