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Source: 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

{\displaystyle K}

is a subset of a real or complex vector space

X
,

{\displaystyle X,}

then the Minkowski functional or gauge of

K

{\displaystyle K}

is defined to be the function

p

K

:
X
→
[
0
,
∞
]
,

{\displaystyle p_{K}:X\to [0,\infty ],}

valued in the extended real numbers, defined by

p

K

(
x
)
:=
inf
{
r
∈

R

:
r
>
0

and

x
∈
r
K
}

for every

x
∈
X
,

{\displaystyle p_{K}(x):=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}\quad {\text{ for every }}x\in X,}

where the infimum of the empty set is defined to be positive infinity

∞

{\displaystyle \,\infty \,}

(which is not a real number so that

p

K

(
x
)

{\displaystyle p_{K}(x)}

would then not be real-valued).
The set

K

{\displaystyle K}

is often assumed/picked to have properties, such as being an absorbing disk in

X
,

{\displaystyle X,}

that guarantee that

p

K

{\displaystyle p_{K}}

will be a real-valued seminorm on

X
.

{\displaystyle X.}

In fact, every seminorm

p

{\displaystyle p}

on

X

{\displaystyle X}

is equal to the Minkowski functional (that is,

p
=

p

K

{\displaystyle p=p_{K}}

) of any subset

K

{\displaystyle K}

of

X

{\displaystyle X}

satisfying

{
x
∈
X
:
p
(
x
)
<
1
}
⊆
K
⊆
{
x
∈
X
:
p
(
x
)
≤
1
}

{\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

X

{\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

X

{\displaystyle X}

into certain algebraic properties of a function on

X
.

{\displaystyle X.}

The Minkowski function is always non-negative (meaning

p

K

≥
0

{\displaystyle 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,

p

K

{\displaystyle p_{K}}

might not be real-valued since for any given

x
∈
X
,

{\displaystyle x\in X,}

the value

p

K

(
x
)

{\displaystyle p_{K}(x)}

is a real number if and only if

{
r
>
0
:
x
∈
r
K
}

{\displaystyle \{r>0:x\in rK\}}

is not empty.
Consequently,

K

{\displaystyle K}

is usually assumed to have properties (such as being absorbing in

X
,

{\displaystyle X,}

for instance) that will guarantee that

p

K

{\displaystyle p_{K}}

is real-valued.

Definition

Minkowski

functional

The products

0
⋅
∞

{\displaystyle 0\cdot \infty }

and

0
⋅
−
∞

{\displaystyle 0\cdot -\infty }

remain undefined.
Some conditions making a gauge real-valued
In the field of convex analysis, the map

p

K

{\displaystyle p_{K}}

taking on the value of

∞

{\displaystyle \,\infty \,}

is not necessarily an issue.
However, in functional analysis

p

K

{\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

{
r
>
0
:
x
∈
r
K
}

{\displaystyle \{r>0:x\in rK\}}

is non-empty for every

x
∈
X
.

{\displaystyle x\in X.}

In order for

p

K

{\displaystyle p_{K}}

to be real-valued, it suffices for the origin of

X

{\displaystyle X}

to belong to the algebraic interior or core of

K

{\displaystyle K}

in

X
.

{\displaystyle X.}

If

K

{\displaystyle K}

is absorbing in

X
,

{\displaystyle X,}

where recall that this implies that

0
∈
K
,

{\displaystyle 0\in K,}

then the origin belongs to the algebraic interior of

K

{\displaystyle K}

in

X

{\displaystyle X}

and thus

p

K

{\displaystyle p_{K}}

is real-valued.
Characterizations of when

p

K

{\displaystyle p_{K}}

is real-valued are given below.

Motivating examples

Example 1
Consider a normed vector space

(
X
,
‖

⋅

‖
)
,

{\displaystyle (X,\|\,\cdot \,\|),}

with the norm

‖

⋅

‖

{\displaystyle \|\,\cdot \,\|}

and let

U
:=
{
x
∈
X
:
‖
x
‖
≤
1
}

{\displaystyle U:=\{x\in X:\|x\|\leq 1\}}

be the unit ball in

X
.

{\displaystyle X.}

Then for every

x
∈
X
,

{\displaystyle x\in X,}

‖
x
‖
=

p

U

(
x
)
.

{\displaystyle \|x\|=p_{U}(x).}

Thus the Minkowski functional

p

U

{\displaystyle p_{U}}

is just the norm on

X
.

{\displaystyle X.}

Example 2
Let

X

{\displaystyle X}

be a vector space without topology with underlying scalar field

K

.

{\displaystyle \mathbb {K} .}

Let

f
:
X
→

K

{\displaystyle f:X\to \mathbb {K} }

be any linear functional on

X

{\displaystyle X}

(not necessarily continuous).
Fix

a
>
0.

{\displaystyle a>0.}

Let

K

{\displaystyle K}

be the set

K
:=
{
x
∈
X
:

|

f
(
x
)

|

≤
a
}

{\displaystyle K:=\{x\in X:|f(x)|\leq a\}}

and let

p

K

{\displaystyle p_{K}}

be the Minkowski functional of

K
.

{\displaystyle K.}

Then

p

K

(
x
)
=

1
a

|

f
(
x
)

|

for all

x
∈
X
.

{\displaystyle p_{K}(x)={\frac {1}{a}}|f(x)|\quad {\text{ for all }}x\in X.}

The function

p

K

{\displaystyle p_{K}}

has the following properties:

It is subadditive:

p

K

(
x
+
y
)
≤

p

K

(
x
)
+

p

K

(
y
)
.

{\displaystyle p_{K}(x+y)\leq p_{K}(x)+p_{K}(y).}

It is absolutely homogeneous:

p

K

(
s
x
)
=

|

s

|

p

K

(
x
)

{\displaystyle p_{K}(sx)=|s|p_{K}(x)}

for all scalars

s
.

{\displaystyle s.}

It is nonnegative:

p

K

≥
0.

{\displaystyle p_{K}\geq 0.}

Therefore,

p

K

{\displaystyle p_{K}}

is a seminorm on

X
,

{\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,

p

K

(
x
)
=
0

{\displaystyle p_{K}(x)=0}

need not imply

x
=
0.

{\displaystyle x=0.}

In the above example, one can take a nonzero

x

{\displaystyle x}

from the kernel of

f
.

{\displaystyle f.}

Consequently, the resulting topology need not be Hausdorff.

Common conditions guaranteeing gauges are seminorms

To guarantee that

p

K

(
0
)
=
0
,

{\displaystyle p_{K}(0)=0,}

it will henceforth be assumed that

0
∈
K
.

{\displaystyle 0\in K.}

In order for

p

K

{\displaystyle p_{K}}

to be a seminorm, it suffices for

K

{\displaystyle K}

to be a disk (that is, convex and balanced) and absorbing in

X
,

{\displaystyle X,}

which are the most common assumption placed on

K
.

{\displaystyle K.}

More generally, if

K

{\displaystyle K}

is convex and the origin belongs to the algebraic interior of

K
,

{\displaystyle K,}

then

p

K

{\displaystyle p_{K}}

is a nonnegative sublinear functional on

X
,

{\displaystyle X,}

which implies in particular that it is subadditive and positive homogeneous.
If

K

{\displaystyle K}

is absorbing in

X

{\displaystyle X}

then

p

[
0
,
1
]
K

{\displaystyle p_{[0,1]K}}

is positive homogeneous, meaning that

p

[
0
,
1
]
K

(
s
x
)
=
s

p

[
0
,
1
]
K

(
x
)

{\displaystyle p_{[0,1]K}(sx)=sp_{[0,1]K}(x)}

for all real

s
≥
0
,

{\displaystyle s\geq 0,}

where

[
0
,
1
]
K
=
{
t
k
:
t
∈
[
0
,
1
]
,
k
∈
K
}
.

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

If

q

{\displaystyle q}

is a nonnegative real-valued function on

X

{\displaystyle X}

that is positive homogeneous, then the sets

U
:=
{
x
∈
X
:
q
(
x
)
<
1
}

{\displaystyle U:=\{x\in X:q(x)<1\}}

and

D
:=
{
x
∈
X
:
q
(
x
)
≤
1
}

{\displaystyle D:=\{x\in X:q(x)\leq 1\}}

satisfy

[
0
,
1
]
U
=
U

{\displaystyle [0,1]U=U}

and

[
0
,
1
]
D
=
D
;

{\displaystyle [0,1]D=D;}

if in addition

q

{\displaystyle q}

is absolutely homogeneous then both

U

{\displaystyle U}

and

D

{\displaystyle D}

are balanced.

= Gauges of absorbing disks

=
Arguably the most common requirements placed on a set

K

{\displaystyle K}

to guarantee that

p

K

{\displaystyle p_{K}}

is a seminorm are that

K

{\displaystyle K}

be an absorbing disk in

X
.

{\displaystyle X.}

Due to how common these assumptions are, the properties of a Minkowski functional

p

K

{\displaystyle p_{K}}

when

K

{\displaystyle K}

is an absorbing disk will now be investigated.
Since all of the results mentioned above made few (if any) assumptions on

K
,

{\displaystyle K,}

they can be applied in this special case.

= Algebraic properties

=
Let

X

{\displaystyle X}

be a real or complex vector space and let

K

{\displaystyle K}

be an absorbing disk in

X
.

{\displaystyle X.}

p

K

{\displaystyle p_{K}}

is a seminorm on

X
.

{\displaystyle X.}

p

K

{\displaystyle p_{K}}

is a norm on

X

{\displaystyle X}

if and only if

K

{\displaystyle K}

does not contain a non-trivial vector subspace.

p

s
K

=

1

|

s

|

p

K

{\displaystyle p_{sK}={\frac {1}{|s|}}p_{K}}

for any scalar

s
≠
0.

{\displaystyle s\neq 0.}

If

J

{\displaystyle J}

is an absorbing disk in

X

{\displaystyle X}

and

J
⊆
K

{\displaystyle J\subseteq K}

then

p

K

≤

p

J

.

{\displaystyle p_{K}\leq p_{J}.}

If

K

{\displaystyle K}

is a set satisfying

{
x
∈
X
:
p
(
x
)
<
1
}

⊆

K

⊆

{
x
∈
X
:
p
(
x
)
≤
1
}

{\displaystyle \{x\in X:p(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p(x)\leq 1\}}

then

K

{\displaystyle K}

is absorbing in

X

{\displaystyle X}

and

p
=

p

K

,

{\displaystyle p=p_{K},}

where

p

K

{\displaystyle p_{K}}

is the Minkowski functional associated with

K
;

{\displaystyle K;}

that is, it is the gauge of

K
.

{\displaystyle K.}

In particular, if

K

{\displaystyle K}

is as above and

q

{\displaystyle q}

is any seminorm on

X
,

{\displaystyle X,}

then

q
=
p

{\displaystyle q=p}

if and only if

{
x
∈
X
:
q
(
x
)
<
1
}

⊆

K

⊆

{
x
∈
X
:
q
(
x
)
≤
1
}
.

{\displaystyle \{x\in X:q(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:q(x)\leq 1\}.}

If

x
∈
X

{\displaystyle x\in X}

satisfies

p

K

(
x
)
<
1

{\displaystyle p_{K}(x)<1}

then

x
∈
K
.

{\displaystyle x\in K.}

= Topological properties

=
Assume that

X

{\displaystyle X}

is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let

K

{\displaystyle K}

be an absorbing disk in

X
.

{\displaystyle X.}

Then

Int

X

⁡
K

⊆

{
x
∈
X
:

p

K

(
x
)
<
1
}

⊆

K

⊆

{
x
∈
X
:

p

K

(
x
)
≤
1
}

⊆

Cl

X

⁡
K
,

{\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

Int

X

⁡
K

{\displaystyle \operatorname {Int} _{X}K}

is the topological interior and

Cl

X

⁡
K

{\displaystyle \operatorname {Cl} _{X}K}

is the topological closure of

K

{\displaystyle K}

in

X
.

{\displaystyle X.}

Importantly, it was not assumed that

p

K

{\displaystyle p_{K}}

was continuous nor was it assumed that

K

{\displaystyle K}

had any topological properties.
Moreover, the Minkowski functional

p

K

{\displaystyle p_{K}}

is continuous if and only if

K

{\displaystyle K}

is a neighborhood of the origin in

X
.

{\displaystyle X.}

If

p

K

{\displaystyle p_{K}}

is continuous then

Int

X

⁡
K
=
{
x
∈
X
:

p

K

(
x
)
<
1
}

and

Cl

X

⁡
K
=
{
x
∈
X
:

p

K

(
x
)
≤
1
}
.

{\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\}.}

Minimal requirements on the set

This section will investigate the most general case of the gauge of any subset

K

{\displaystyle K}

of

X
.

{\displaystyle X.}

The more common special case where

K

{\displaystyle K}

is assumed to be an absorbing disk in

X

{\displaystyle X}

was discussed above.

= Properties

=
All results in this section may be applied to the case where

K

{\displaystyle K}

is an absorbing disk.
Throughout,

K

{\displaystyle K}

is any subset of

X
.

{\displaystyle X.}

= Examples

=

If

L

{\displaystyle {\mathcal {L}}}

is a non-empty collection of subsets of

X

{\displaystyle X}

then

p

∪

L

(
x
)
=
inf

{

p

L

(
x
)
:
L
∈

L

}

{\displaystyle p_{\cup {\mathcal {L}}}(x)=\inf \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}

for all

x
∈
X
,

{\displaystyle x\in X,}

where

∪

L

=

def

⋃

L
∈

L

L
.

{\displaystyle \cup {\mathcal {L}}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~{\textstyle \bigcup \limits _{L\in {\mathcal {L}}}}L.}

Thus

p

K
∪
L

(
x
)
=
min

{

p

K

(
x
)
,

p

L

(
x
)

}

{\displaystyle p_{K\cup L}(x)=\min \left\{p_{K}(x),p_{L}(x)\right\}}

for all

x
∈
X
.

{\displaystyle x\in X.}

If

L

{\displaystyle {\mathcal {L}}}

is a non-empty collection of subsets of

X

{\displaystyle X}

and

I
⊆
X

{\displaystyle I\subseteq X}

satisfies

{

x
∈
X
:

p

L

(
x
)
<
1

for all

L
∈

L

}

⊆

I

⊆

{

x
∈
X
:

p

L

(
x
)
≤
1

for all

L
∈

L

}

{\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

p

I

(
x
)
=
sup

{

p

L

(
x
)
:
L
∈

L

}

{\displaystyle p_{I}(x)=\sup \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}

for all

x
∈
X
.

{\displaystyle x\in X.}

The following examples show that the containment

(
0
,
R
]
K

⊆

⋂

e
>
0

(
0
,
R
+
e
)
K

{\displaystyle (0,R]K\;\subseteq \;{\textstyle \bigcap \limits _{e>0}}(0,R+e)K}

could be proper.
Example: If

R
=
0

{\displaystyle R=0}

and

K
=
X

{\displaystyle K=X}

then

(
0
,
R
]
K
=
(
0
,
0
]
X
=
∅
X
=
∅

{\displaystyle (0,R]K=(0,0]X=\varnothing X=\varnothing }

but

⋂

e
>
0

(
0
,
e
)
K
=

⋂

e
>
0

X
=
X
,

{\displaystyle {\textstyle \bigcap \limits _{e>0}}(0,e)K={\textstyle \bigcap \limits _{e>0}}X=X,}

which shows that its possible for

(
0
,
R
]
K

{\displaystyle (0,R]K}

to be a proper subset of

⋂

e
>
0

(
0
,
R
+
e
)
K

{\displaystyle {\textstyle \bigcap \limits _{e>0}}(0,R+e)K}

when

R
=
0.

{\displaystyle R=0.}

◼

{\displaystyle \blacksquare }

The next example shows that the containment can be proper when

R
=
1
;

{\displaystyle R=1;}

the example may be generalized to any real

R
>
0.

{\displaystyle R>0.}

Assuming that

[
0
,
1
]
K
⊆
K
,

{\displaystyle [0,1]K\subseteq K,}

the following example is representative of how it happens that

x
∈
X

{\displaystyle x\in X}

satisfies

p

K

(
x
)
=
1

{\displaystyle p_{K}(x)=1}

but

x
∉
(
0
,
1
]
K
.

{\displaystyle x\not \in (0,1]K.}

Example: Let

x
∈
X

{\displaystyle x\in X}

be non-zero and let

K
=
[
0
,
1
)
x

{\displaystyle K=[0,1)x}

so that

[
0
,
1
]
K
=
K

{\displaystyle [0,1]K=K}

and

x
∉
K
.

{\displaystyle x\not \in K.}

From

x
∉
(
0
,
1
)
K
=
K

{\displaystyle x\not \in (0,1)K=K}

it follows that

p

K

(
x
)
≥
1.

{\displaystyle p_{K}(x)\geq 1.}

That

p

K

(
x
)
≤
1

{\displaystyle p_{K}(x)\leq 1}

follows from observing that for every

e
>
0
,

{\displaystyle e>0,}

(
0
,
1
+
e
)
K
=
[
0
,
1
+
e
)
(
[
0
,
1
)
x
)
=
[
0
,
1
+
e
)
x
,

{\displaystyle (0,1+e)K=[0,1+e)([0,1)x)=[0,1+e)x,}

which contains

x
.

{\displaystyle x.}

Thus

p

K

(
x
)
=
1

{\displaystyle p_{K}(x)=1}

and

x
∈

⋂

e
>
0

(
0
,
1
+
e
)
K
.

{\displaystyle x\in {\textstyle \bigcap \limits _{e>0}}(0,1+e)K.}

However,

(
0
,
1
]
K
=
(
0
,
1
]
(
[
0
,
1
)
x
)
=
[
0
,
1
)
x
=
K

{\displaystyle (0,1]K=(0,1]([0,1)x)=[0,1)x=K}

so that

x
∉
(
0
,
1
]
K
,

{\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

f
:
X
→
[
0
,
∞
]

{\displaystyle f:X\to [0,\infty ]}

that have a certain purely algebraic property that is commonly encountered.

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

f
:
X
→

R

{\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 on star sets

= Characterizing Minkowski functionals that are seminorms

=
In this next theorem, which follows immediately from the statements above,

K

{\displaystyle K}

is not assumed to be absorbing in

X

{\displaystyle X}

and instead, it is deduced that

(
0
,
1
)
K

{\displaystyle (0,1)K}

is absorbing when

p

K

{\displaystyle p_{K}}

is a seminorm. It is also not assumed that

K

{\displaystyle K}

is balanced (which is a property that

K

{\displaystyle K}

is often required to have); in its place is the weaker condition that

(
0
,
1
)
s
K
⊆
(
0
,
1
)
K

{\displaystyle (0,1)sK\subseteq (0,1)K}

for all scalars

s

{\displaystyle s}

satisfying

|

s

|

=
1.

{\displaystyle |s|=1.}

The common requirement that

K

{\displaystyle K}

be convex is also weakened to only requiring that

(
0
,
1
)
K

{\displaystyle (0,1)K}

be convex.

= Positive sublinear functions and Minkowski functionals

=
It may be shown that a real-valued subadditive function

f
:
X
→

R

{\displaystyle f:X\to \mathbb {R} }

on an arbitrary topological vector space

X

{\displaystyle X}

is continuous at the origin if and only if it is uniformly continuous, where if in addition

f

{\displaystyle f}

is nonnegative, then

f

{\displaystyle f}

is continuous if and only if

V
:=
{
x
∈
X
:
f
(
x
)
<
1
}

{\displaystyle V:=\{x\in X:f(x)<1\}}

is an open neighborhood in

X
.

{\displaystyle X.}

If

f
:
X
→

R

{\displaystyle f:X\to \mathbb {R} }

is subadditive and satisfies

f
(
0
)
=
0
,

{\displaystyle f(0)=0,}

then

f

{\displaystyle f}

is continuous if and only if its absolute value

|

f

|

:
X
→
[
0
,
∞
)

{\displaystyle |f|:X\to [0,\infty )}

is continuous.
A nonnegative sublinear function is a nonnegative homogeneous function

f
:
X
→
[
0
,
∞
)

{\displaystyle f:X\to [0,\infty )}

that satisfies the triangle inequality.
It follows immediately from the results below that for such a function

f
,

{\displaystyle f,}

if

V
:=
{
x
∈
X
:
f
(
x
)
<
1
}

{\displaystyle V:=\{x\in X:f(x)<1\}}

then

f
=

p

V

.

{\displaystyle f=p_{V}.}

Given

K
⊆
X
,

{\displaystyle K\subseteq X,}

the Minkowski functional

p

K

{\displaystyle p_{K}}

is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if

(
0
,
∞
)
K
=
X

{\displaystyle (0,\infty )K=X}

and

(
0
,
1
)
K

{\displaystyle (0,1)K}

is convex.
Correspondence between open convex sets and positive continuous sublinear functions

Notes

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