Minkowski functional

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 $${\displaystyle p_{K}(x):=\inf\{r>0:x\in rK\},}$$  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 ${\displaystyle X.}$ ^[1] 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.

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 $${\displaystyle K:=\{x\in X:|f(x)|\leq a\}}$$  and let ${\displaystyle p_{K}}$  be the Minkowski functional of ${\displaystyle K.}$  Then $${\displaystyle p_{K}(x)={\frac {1}{a}}|f(x)|\quad {\text{ for all }}x\in X.}$$  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 absolutely 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.

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 ${\displaystyle [0,1]K=\{tk:t\in [0,1],k\in K\}.}$ ^[3] 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.^[3]

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.

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.^[4]
• ${\displaystyle p_{sK}={\frac {1}{|s|}}p_{K}}$  for any scalar ${\displaystyle s\neq 0.}$ ^[4]
• 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 ${\displaystyle K.}$ ^[5]
  • 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 ${\displaystyle \{x\in X:q(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:q(x)\leq 1\}.}$ ^[5]
• If ${\displaystyle x\in X}$  satisfies ${\displaystyle p_{K}(x)<1}$  then ${\displaystyle x\in K.}$ 

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 $${\displaystyle \operatorname {Int} _{X}K\;\subseteq \;\{x\in X:p_{K}(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p_{K}(x)\leq 1\}\;\subseteq \;\operatorname {Cl} _{X}K,}$$  where ${\displaystyle \operatorname {Int} _{X}K}$  is the topological interior and ${\displaystyle \operatorname {Cl} _{X}K}$  is the topological closure of ${\displaystyle K}$  in ${\displaystyle X.}$ ^[6] 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 ${\displaystyle X.}$ ^[6] If ${\displaystyle p_{K}}$  is continuous then^[6] $${\displaystyle \operatorname {Int} _{X}K=\{x\in X:p_{K}(x)<1\}\quad {\text{ and }}\quad \operatorname {Cl} _{X}K=\{x\in X:p_{K}(x)\leq 1\}.}$$ 

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

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 ${\displaystyle X.}$ 

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 ${\displaystyle \cup {\mathcal {L}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \bigcup \limits _{L\in {\mathcal {L}}}}L.}$ 
   • 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 ${\displaystyle (0,R]K\;\subseteq \;{\textstyle \bigcap \limits _{e>0}}(0,R+e)K}$  could be proper.

Example: If ${\displaystyle R=0}$  and ${\displaystyle K=X}$  then ${\displaystyle (0,R]K=(0,0]X=\varnothing X=\varnothing }$  but ${\displaystyle {\textstyle \bigcap \limits _{e>0}}(0,e)K={\textstyle \bigcap \limits _{e>0}}X=X,}$  which shows that its possible for ${\displaystyle (0,R]K}$  to be a proper subset of ${\displaystyle {\textstyle \bigcap \limits _{e>0}}(0,R+e)K}$  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 ${\displaystyle x\in {\textstyle \bigcap \limits _{e>0}}(0,1+e)K.}$  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. ${\displaystyle \blacksquare }$ 

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 commonly encountered.