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{{Short description|Function made from a set}}
[[File:Minkowski functional.svg|thumb]]

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 <math>K</math> is a subset of a [[Real number|real]] or [[Complex number|complex]] [[vector space]] <math>X,</math> then the {{em|Minkowski functional}} or {{em|gauge}} of <math>K</math> is defined to be the [[Function (mathematics)|function]] <math>p_K : X \to [0, \infty],</math> valued in the [[Extended real number line|extended real number]]s, defined by
<math display=block>p_K(x) := \inf \{r \in \R : r > 0 \text{ and } x \in r K\} \quad \text{ for every } x \in X,</math>
where the [[Infimum and supremum|infimum]] of the empty set is defined to be [[positive infinity]] <math>\,\infty\,</math> (which is {{em|not}} a real number so that <math>p_K(x)</math> would then {{em|not}} be real-valued). 

The set <math>K</math> is often assumed/picked to have properties, such as being an absorbing [[Absolutely convex set|disk]] in <math>X,</math> that guarantee that <math>p_K</math> will be a real-valued [[seminorm]] on <math>X.</math> 
In fact, every seminorm <math>p</math> on <math>X</math> is equal to the Minkowski functional (that is, <math>p = p_K</math>) of any subset <math>K</math> of <math>X</math> satisfying <math>\{x \in X : p(x) < 1\} \subseteq K \subseteq \{x \in X : p(x) \leq 1\}</math> (where all three of these sets are necessarily absorbing in <math>X</math> and the first and last are also disks). 

Thus every seminorm (which is a {{em|function}} defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a {{em|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 {{em|geometric}} properties of a subset of <math>X</math> into certain {{em|algebraic}} properties of a function on <math>X.</math> 

The Minkowski function is always non-negative (meaning <math>p_K \geq 0</math>). 
This property of being nonnegative stands in contrast to other classes of functions, such as [[sublinear function]]s and real [[linear functionals]], that do allow negative values. 
However, <math>p_K</math> might not be real-valued since for any given <math>x \in X,</math> the value <math>p_K(x)</math> is a real number if and only if <math>\{r > 0 : x \in r K\}</math> is not [[Empty set|empty]].
Consequently, <math>K</math> is usually assumed to have properties (such as being [[Absorbing set|absorbing]] in <math>X,</math> for instance) that will guarantee that <math>p_K</math> is real-valued. 

==Definition==

Let <math>K</math> be a subset of a real or complex vector space <math>X.</math> Define the {{em|gauge}} of <math>K</math> or the {{em|Minkowski functional}} associated with or induced by <math>K</math> as being the function <math>p_K : X \to [0, \infty],</math> valued in the [[Extended real number line|extended real number]]s, defined by 
<math display=block>p_K(x) := \inf \{r > 0 : x \in r K\},</math> 
where recall that the [[infimum]] of the empty set is <math>\,\infty\,</math> (that is, <math>\inf \varnothing = \infty</math>). Here, <math>\{r > 0 : x \in r K\}</math> is shorthand for <math>\{r \in \R : r > 0 \text{ and } x \in r K\}.</math> 

For any <math>x \in X,</math> <math>p_K(x) \neq \infty</math> if and only if <math>\{r > 0 : x \in r K\}</math> is not empty. 
The arithmetic operations on <math>\R</math> [[Extended real number line#Arithmetic operations|can be extended]] to operate on <math>\pm \infty,</math> where <math>\frac{r}{\pm \infty} := 0</math> for all non-zero real <math>- \infty < r < \infty.</math> 
The products <math>0 \cdot \infty</math> and <math>0 \cdot - \infty</math> remain undefined.

'''Some conditions making a gauge real-valued'''

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

In order for <math>p_K</math> to be real-valued, it suffices for the origin of <math>X</math> to belong to the {{em|[[algebraic interior]]}} or {{em|core}} of <math>K</math> in <math>X.</math>{{sfn|Narici|Beckenstein|2011|p=109}} 
If <math>K</math> is [[Absorbing set|absorbing]] in <math>X,</math> where recall that this implies that <math>0 \in K,</math> then the origin belongs to the [[algebraic interior]] of <math>K</math> in <math>X</math> and thus <math>p_K</math> is real-valued. 
Characterizations of when <math>p_K</math> is real-valued are given below.

==Motivating examples==

'''Example 1'''

Consider a [[normed vector space]] <math>(X, \|\,\cdot\,\|),</math> with the norm <math>\|\,\cdot\,\|</math> and let <math>U := \{x\in X : \|x\| \leq 1\}</math> be the unit ball in <math>X.</math> Then for every <math>x \in X,</math> <math>\|x\| = p_U(x).</math> Thus the Minkowski functional <math>p_U</math> is just the norm on <math>X.</math> 

'''Example 2'''

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

#It is {{em|subadditive}}: <math>p_K(x + y) \leq p_K(x) + p_K(y).</math>
#It is {{em|absolutely homogeneous}}: <math>p_K(s x) = |s| p_K(x)</math> for all scalars <math>s.</math>
#It is {{em|nonnegative}}: <math>p_K \geq 0.</math> 

Therefore, <math>p_K</math> is a [[seminorm]] on <math>X,</math> 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, <math>p_K(x) = 0</math> need not imply <math>x = 0.</math> 
In the above example, one can take a nonzero <math>x</math> from the kernel of <math>f.</math> 
Consequently, the resulting topology need not be [[Hausdorff space|Hausdorff]].

==Common conditions guaranteeing gauges are seminorms==

To guarantee that <math>p_K(0) = 0,</math> it will henceforth be assumed that <math>0 \in K.</math>

In order for <math>p_K</math> to be a seminorm, it suffices for <math>K</math> to be a [[Absolutely convex set|disk]] (that is, convex and balanced) and absorbing in <math>X,</math> which are the most common assumption placed on <math>K.</math>

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|p=119}}|math_statement=
If <math>K</math> is an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in a vector space <math>X</math> then the Minkowski functional of <math>K,</math> which is the map <math>p_K : X \to [0, \infty)</math> defined by
<math display=block>p_K(x) := \inf \{r > 0 : x \in r K\},</math>
is a seminorm on <math>X.</math>
Moreover,
<math display=block>p_K(x) = \frac{1}{\sup \{r > 0 : r x \in K\}}.</math>
}}

More generally, if <math>K</math> is convex and the origin belongs to the [[algebraic interior]] of <math>K,</math> then <math>p_K</math> is a nonnegative [[sublinear functional]] on <math>X,</math> which implies in particular that it is [[subadditive]] and [[Positive homogeneity|positive homogeneous]].
If <math>K</math> is absorbing in <math>X</math> then <math>p_{[0, 1] K}</math> is positive homogeneous, meaning that <math>p_{[0, 1] K}(s x) = s p_{[0, 1] K}(x)</math> for all real <math>s \geq 0,</math> where <math>[0, 1] K = \{t k : t \in [0, 1], k \in K\}.</math>{{sfn|Jarchow|1981|pp=104-108}}
If <math>q</math> is a nonnegative real-valued function on <math>X</math> that is positive homogeneous, then the sets <math>U := \{x \in X : q(x) < 1\}</math> and <math>D := \{x \in X : q(x) \leq 1\}</math> satisfy <math>[0, 1] U = U</math> and <math>[0, 1] D = D;</math> 
if in addition <math>q</math> is absolutely homogeneous then both <math>U</math> and <math>D</math> are [[Balanced set|balanced]].{{sfn|Jarchow|1981|pp=104-108}} 

===Gauges of absorbing disks===

Arguably the most common requirements placed on a set <math>K</math> to guarantee that <math>p_K</math> is a seminorm are that <math>K</math> be an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in <math>X.</math>
Due to how common these assumptions are, the properties of a Minkowski functional <math>p_K</math> when <math>K</math> is an absorbing disk will now be investigated. 
Since all of the results mentioned above made few (if any) assumptions on <math>K,</math> they can be applied in this special case. 

{{Math theorem|name=Theorem|math_statement=
Assume that <math>K</math> is an absorbing subset of <math>X.</math>
It is shown that: 
# If <math>K</math> is [[Convex set|convex]] then <math>p_K</math> is subadditive.
# If <math>K</math> is [[Balanced set|balanced]] then <math>p_K</math> is [[Homogeneous function|absolutely homogeneous]]; that is, <math>p_K(s x) = |s| p_K(x)</math> for all scalars <math>s.</math>
}}

{{collapse top|title=Proof that the Gauge of an absorbing disk is a seminorm|left=true}}
'''Convexity and subadditivity'''

A simple geometric argument that shows convexity of <math>K</math> implies subadditivity is as follows. 
Suppose for the moment that <math>p_K(x) = p_K(y) = r.</math> 
Then for all <math>e > 0,</math> <math>x, y \in K_e := (r, e) K.</math> 
Since <math>K</math> is convex and <math>r + e \neq 0,</math> <math>K_e</math> is also convex. 
Therefore, <math>\frac{1}{2} x + \frac{1}{2} y \in K_e.</math> 
By definition of the Minkowski functional <math>p_K,</math> 
<math display=block>p_K\left(\frac{1}{2} x + \frac{1}{2} y\right) \leq r + e = \frac{1}{2} p_K(x) + \frac{1}{2} p_K(y) + e.</math>

But the left hand side is <math>\frac{1}{2} p_K(x + y),</math> so that 
<math display=block>p_K(x + y) \leq p_K(x) + p_K(y) + 2 e.</math>

Since <math>e > 0</math> was arbitrary, it follows that <math>p_K(x + y) \leq p_K(x) + p_K(y),</math> which is the desired inequality. 
The general case <math>p_K(x) > p_K(y)</math> is obtained after the obvious modification.

Convexity of <math>K,</math> together with the initial assumption that the set <math>\{r > 0 : x \in r K\}</math> is nonempty, implies that <math>K</math> is [[Absorbing set|absorbing]].

'''Balancedness and absolute homogeneity'''

Notice that <math>K</math> being balanced implies that
<math display=block>\lambda x \in r K \quad \mbox{if and only if} \quad x \in \frac{r}{|\lambda|} K.</math>

Therefore
<math display=block>p_K (\lambda x) = \inf \left\{r > 0 : \lambda x \in r K \right\} 
= \inf \left\{r > 0 : x \in \frac{r}{|\lambda|} K \right\}
= \inf \left\{|\lambda|\frac{r}{|\lambda|} > 0 : x \in \frac{r}{|\lambda|} K \right\}
= |\lambda| p_K(x).
</math>
{{collapse bottom}}

===Algebraic properties===

Let <math>X</math> be a real or complex vector space and let <math>K</math> be an absorbing disk in <math>X.</math> 

<ul>
<li><math>p_K</math> is a [[seminorm]] on <math>X.</math></li>
<li><math>p_K</math> is a [[Norm (mathematics)|norm]] on <math>X</math> if and only if <math>K</math> does not contain a non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|pp=115-154}}</li>
<li><math>p_{s K} = \frac{1}{|s|} p_K</math> for any scalar <math>s \neq 0.</math>{{sfn|Narici|Beckenstein|2011|pp=115-154}}</li>
<li>If <math>J</math> is an absorbing disk in <math>X</math> and <math>J \subseteq K</math> then <math>p_K \leq p_J.</math></li>
<li>If <math>K</math> is a set satisfying <math>\{x \in X : p(x) < 1\} \; \subseteq \; K \; \subseteq \; \{x \in X : p(x) \leq 1\}</math> then <math>K</math> is [[Absorbing set|absorbing]] in <math>X</math> and <math>p = p_K,</math> where <math>p_K</math> is the Minkowski functional associated with <math>K;</math> that is, it is the gauge of <math>K.</math>{{sfn|Schaefer|1999|p=40}}
* In particular, if <math>K</math> is as above and <math>q</math> is any seminorm on <math>X,</math> then <math>q = p</math> if and only if <math>\{x \in X : q(x) < 1\} \; \subseteq \; K \; \subseteq \; \{x \in X : q(x) \leq 1\}.</math>{{sfn|Schaefer|1999|p=40}}</li>
<li>If <math>x \in X</math> satisfies <math>p_K(x) < 1</math> then <math>x \in K.</math></li>
</ul>

===Topological properties===

Assume that <math>X</math> is a (real or complex) [[topological vector space]] (TVS) (not necessarily [[Hausdorff space|Hausdorff]] or [[locally convex]]) and let <math>K</math> be an absorbing disk in <math>X.</math> Then
<math display=block>\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,</math>
where <math>\operatorname{Int}_X K</math>  is the [[topological interior]] and <math>\operatorname{Cl}_X K</math> is the [[Closure (topology)|topological closure]] of <math>K</math> in <math>X.</math>{{sfn|Narici|Beckenstein|2011|p=119-120}} 
Importantly, it was {{em|not}} assumed that <math>p_K</math> was continuous nor was it assumed that <math>K</math> had any topological properties. 

Moreover, the Minkowski functional <math>p_K</math> is continuous if and only if <math>K</math> is a neighborhood of the origin in <math>X.</math>{{sfn|Narici|Beckenstein|2011|p=119-120}} 
If <math>p_K</math> is continuous then{{sfn|Narici|Beckenstein|2011|p=119-120}}
<math display=block>\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\}.</math>

==Minimal requirements on the set==

This section will investigate the most general case of the gauge of {{em|any}} subset <math>K</math> of <math>X.</math> 
The more common special case where <math>K</math> is assumed to be an [[Absorbing set|absorbing]] [[Absolutely convex set|disk]] in <math>X</math> was discussed above. 

===Properties===

All results in this section may be applied to the case where <math>K</math> is an absorbing disk. 

Throughout, <math>K</math> is any subset of <math>X.</math>

{{Math theorem|name=Summary|math_statement=
Suppose that <math>K</math> is a subset of a real or complex vector space <math>X.</math> 
<ol>
<li>'''[[Strict positive homogeneity]]''': <math>p_K(r x) = r p_K(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>r > 0.</math>
* '''[[Nonnegative homogeneity|Positive/Nonnegative homogeneity]]''': <math>p_K</math> is nonnegative homogeneous if and only if <math>p_K</math> is real-valued.
** A map <math>p</math> is called {{em|[[Nonnegative homogeneity|nonnegative homogeneous]]}}{{sfn|Kubrusly|2011|p=200}} if <math>p(r x) = r p(x)</math> for all <math>x \in X</math> and all {{em|nonnegative}} real <math>r \geq 0.</math> Since <math>0 \cdot \infty</math> is undefined, a map that takes infinity as a value is not nonnegative homogeneous.</li>
<li>'''Real-values''': <math>(0, \infty) K</math> is the set of all points on which <math>p_K</math> is real valued. So <math>p_K</math> is real-valued if and only if <math>(0, \infty) K = X,</math> in which case <math>0 \in K.</math>
* '''Value at <math>0</math>''': <math>p_K(0) \neq \infty</math> if and only if <math>0 \in K</math> if and only if <math>p_K(0) = 0.</math>
* '''[[Null space]]''': If <math>x \in X</math> then <math>p_K(x) = 0</math> if and only if <math>(0, \infty) x \subseteq (0, 1) K</math> if and only if there exists a divergent sequence of positive real numbers <math>t_1, t_2, t_3, \cdots \to \infty</math> such that <math>t_n x \in K</math> for all <math>n.</math> Moreover, the [[zero set]] of <math>p_K</math> is <math>\ker p_K ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{y \in X : p_K(y) = 0 \right\} = {\textstyle\bigcap\limits_{e > 0}} (0, e) K.</math>
</li>
<li>'''Comparison to a constant''': If <math>0 \leq r \leq \infty</math> then for any <math>x \in X,</math> <math>p_K(x) < r</math> if and only if <math>x \in (0, r) K;</math> this can be restated as: If <math>0 \leq r \leq \infty</math> then <math>p_K^{-1}([0, r)) = (0, r) K.</math>
* It follows that if <math>0 \leq R < \infty</math> is real then <math>p_K^{-1}([0, R]) = {\textstyle\bigcap\limits_{e > 0}} (0, R + e) K,</math> where the set on the right hand side denotes <math>{\textstyle\bigcap\limits_{e > 0}} [(0, R + e) K]</math> and not its subset <math>\left[{\textstyle\bigcap\limits_{e > 0}} (0, R + e)\right] K = (0, R] K.</math> If <math>R > 0</math> then these sets are equal if and only if <math>K</math> contains <math>\left\{y \in X : p_K(y) = 1 \right\}.</math> 
* In particular, if <math>x \in R K</math> or <math>x \in (0, R] K</math> then <math>p_K(x) \leq R,</math> but importantly, the converse is not necessarily true. <!--- Note that if <math>R = 0</math> then <math>(0, R] K = \varnothing</math> so <math>x \in (0, R] K</math> implies <math>R \neq 0.</math> ---></li>
<li>'''Gauge comparison''': For any subset <math>L \subseteq X,</math> <math>p_K \leq p_L</math> if and only if <math>(0, 1) L \subseteq (0, 1) K;</math> thus <math>p_L = p_K</math> if and only if <math>(0, 1) L = (0, 1) K.</math> 
* The assignment <math>L \mapsto p_L</math> is order-reversing in the sense that if <math>K \subseteq L</math> then <math>p_L \leq p_K.</math>{{sfn|Schechter|1996|p=316}}
* Because the set <math>L := (0, 1) K</math> satisfies <math>(0, 1) L = (0, 1) K,</math> it follows that replacing <math>K</math> with <math>p_K^{-1}([0, 1)) = (0, 1) K</math> will not change the resulting Minkowski functional. The same is true of <math>L := (0, 1] K</math> and of <math>L := p_K^{-1}([0, 1]).</math>
* If <math>D ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{y \in X : p_K(y) = 1 \text{ or } p_K(y) = 0 \right\}</math> then <math>p_D = p_K</math> and <math>D</math> has the particularly nice property that if <math>r > 0</math> is real then <math>x \in r D</math> if and only if <math>p_D(x) = r</math> or <math>p_D(x) = 0.</math><ref group=note>It is in general '''{{em|false}}''' that <math>x \in r D</math> if and only if <math>p_D(x) = r</math> (for example, consider when <math>p_K</math> is a [[Norm (mathematics)|norm]] or a seminorm). The correct statement is: If <math>0 < r < \infty</math> then <math>x \in r D</math> if and only if <math>p_D(x) = r</math> or <math>p_D(x) = 0.</math></ref> Moreover, if <math>r > 0</math> is real then <math>p_D(x) \leq r</math> if and only if <math>x \in (0, r] D.</math>
</li>
<li>'''[[Subadditive]]'''/'''Triangle inequality''': <math>p_K</math> is subadditive if and only if <math>(0, 1) K</math> is convex. If <math>K</math> is convex then so are both <math>(0, 1) K</math> and <math>(0, 1] K</math> and moreover, <math>p_K</math> is subadditive.</li>
<li>'''Scaling the set''': If <math>s \neq 0</math> is a scalar then <math>p_{s K}(y) = p_K\left(\tfrac{1}{s} y\right)</math> for all <math>y \in X.</math> 
Thus if <math>0 < r < \infty</math> is real then <math>p_{r K}(y) = p_K\left(\tfrac{1}{r} y\right) = \tfrac{1}{r} p_K(y).</math></li>
<li>'''Symmetric''': <math>p_K</math> is symmetric (meaning that <math>p_K(- y) = p_K(y)</math> for all <math>y \in X</math>) if and only if <math>(0, 1) K</math> is a [[symmetric set]] (meaning that<math>(0, 1) K = - (0, 1) K</math>), which happens if and only if <math>p_K = p_{- K}.</math></li>
<li>'''[[Absolute homogeneity]]''': <math>p_K(u x) = p_K(x)</math> for all <math>x \in X</math> and all unit length scalars <math>u</math><ref group=note><math>u</math> is having unit length means that <math>|u| = 1.</math></ref> if and only if <math>(0, 1) u K \subseteq (0, 1) K</math> for all unit length scalars <math>u,</math> in which case <math>p_K(s x) = |s| p_K(x)</math> for all <math>x \in X</math> and all {{em|non-zero}} scalars <math>s \neq 0.</math> If in addition <math>p_K</math> is also real-valued then this holds for {{em|all}} scalars <math>s</math> (that is, <math>p_K</math> is absolutely homogeneous<ref group=note>The map <math>p_K</math> is called {{em|absolutely homogeneous}} if <math>|s| p_K(x)</math> is well-defined and <math>p_K(s x) = |s| p_K(x)</math> for all <math>x \in X</math> and all scalars <math>s</math> (not just non-zero scalars).</ref>).
* <math>(0, 1) u K \subseteq (0, 1) K</math> for all unit length <math>u</math> if and only if <math>(0, 1) u K = (0, 1) K</math> for all unit length <math>u.</math>
* <math>s K \subseteq K</math> for all unit scalars <math>s</math> if and only if <math>s K = K</math> for all unit scalars <math>s;</math> if this is the case then <math>(0, 1) K = (0, 1) s K</math> for all unit scalars <math>s.</math>
* The Minkowski functional of any [[balanced set]] is a [[balanced function]].{{sfn|Schechter|1996|p=316}} 
</li>
<li>'''[[Absorbing set|Absorbing]]''': If <math>K</math> is convex {{em|or}} balanced and if <math>(0, \infty) K = X</math> then <math>K</math> is absorbing in <math>X.</math>
* If a set <math>A</math> is absorbing in <math>X</math> and <math>A \subseteq K</math> then <math>K</math> is absorbing in <math>X.</math>
* If <math>K</math> is convex and <math>0 \in K</math> then <math>[0, 1] K = K,</math> in which case <math>(0, 1) K \subseteq K.</math></li>
<li>'''Restriction to a vector subspace''': If <math>S</math> is a vector subspace of <math>X</math> and if <math>p_{K \cap S} : S \to [0, \infty]</math> denotes the Minkowski functional of <math>K \cap S</math> on <math>S,</math> then <math>p_K\big\vert_S = p_{K \cap S},</math> where <math>p_K\big\vert_S</math> denotes the [[Restriction of a function|restriction]] of <math>p_K</math> to <math>S.</math></li>
</ol>
}}

{{collapse top|title=Proof|left=true}}
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 <math>A \subseteq X</math> that satisfies <math>(0, \infty) A = X</math> is necessarily [[Absorbing set|absorbing]] in <math>X</math> is straightforward and can be found in the article on [[absorbing set]]s. 

For any real <math>t > 0,</math> 
<math display=block>\{r > 0 : t x \in r K\} = \{t(r/t) : x \in (r/t) K\} = t \{s > 0 : x \in s K\}</math>
so that taking the infimum of both sides shows that 
<math display=block>p_K(tx) = \inf \{r > 0 : t x \in r K\} = t \inf \{s > 0 : x \in s K\} = t p_K(x).</math>
This proves that Minkowski functionals are strictly positive homogeneous. For <math>0 \cdot p_K(x)</math> to be well-defined, it is necessary and sufficient that <math>p_K(x) \neq \infty;</math> thus <math>p_K(tx) = t p_K(x)</math> for all <math>x \in X</math> and all {{em|non-negative}} real <math>t \geq 0</math> if and only if <math>p_K</math> is real-valued. 

The hypothesis of statement (7) allows us to conclude that <math>p_K(s x) = p_K(x)</math> for all <math>x \in X</math> and all scalars <math>s</math> satisfying <math>|s| = 1.</math> 
Every scalar <math>s</math> is of the form <math>r e^{i t}</math> for some real <math>t</math> where <math>r := |s| \geq 0</math> and <math>e^{i t}</math> is real if and only if <math>s</math> is real. 
The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of <math>p_K,</math> and from the positive homogeneity of <math>p_K</math> when <math>p_K</math> is real-valued. 
<math>\blacksquare</math>
{{collapse bottom}}

===Examples===

<ol>
<li>If <math>\mathcal{L}</math> is a non-empty collection of subsets of <math>X</math> then <math>p_{\cup \mathcal{L}}(x) = \inf \left\{p_L(x) : L \in \mathcal{L} \right\}</math> for all <math>x \in X,</math> where <math>\cup \mathcal{L} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ {\textstyle\bigcup\limits_{L \in \mathcal{L}}} L.</math> 
* Thus <math>p_{K \cup L}(x) = \min \left\{p_K(x), p_L(x) \right\}</math> for all <math>x \in X.</math> 
</li>
<li>If <math>\mathcal{L}</math> is a non-empty collection of subsets of <math>X</math> and <math>I \subseteq X</math> satisfies 
<math display=block>\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\}</math> 
then <math>p_I(x) = \sup \left\{p_L(x) : L \in \mathcal{L}\right\}</math> for all <math>x \in X.</math>
</li>
</ol>

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

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

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

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

===Positive homogeneity characterizes Minkowski functionals===

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

{{Math theorem|name=Theorem|math_statement=
Let <math>f : X \to [0, \infty]</math> be any function. 
The following statements are equivalent: 
<ol>
<li>'''[[Strict positive homogeneity]]''': <math>\; f(t x) = t f(x)</math> for all <math>x \in X</math> and all {{em|positive}} real <math>t > 0.</math>
* This statement is equivalent to: <math>f(t x) \leq t f(x)</math> for all <math>x \in X</math> and all positive real <math>t > 0.</math></li>
<li><math>f</math> is a Minkowski functional: meaning that there exists a subset <math>S \subseteq X</math> such that <math>f = p_S.</math></li>
<li><math>f = p_K</math> where <math>K := \{x \in X : f(x) \leq 1\}.</math></li>
<li><math>f = p_V \,</math> where <math>V \,:= \{x \in X : f(x) < 1\}.</math></li>
</ol>
Moreover, if <math>f</math> never takes on the value <math>\,\infty\,</math> (so that the product <math>0 \cdot f(x)</math> is always well-defined) then this list may be extended to include: 
<ol start=5>
<li>'''[[Positive homogeneity|Positive]]'''/'''[[Nonnegative homogeneity]]''': <math>f(t x) = t f(x)</math> for all <math>x \in X</math> and all {{em|nonnegative}} real <math>t \geq 0.</math></li>
</ol>
}}

{{collapse top|title=Proof|left=true}}
If <math>f(t x) \leq t f(x)</math> holds for all <math>x \in X</math> and real <math>t > 0</math> then <math>t f(x) = t f\left(\tfrac{1}{t}(t x)\right) \leq t \tfrac{1}{t} f(t x) = f(t x) \leq t f(x)</math> so that <math>t f(x) = f(t x).</math>

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 <math>f : X \to [0, \infty]</math> is a function such that <math>f(t x) = t f(x)</math> for all <math>x \in X</math> and all real <math>t > 0</math> and let <math>K := \{y \in X : f(y) \leq 1\}.</math> 
<!--- START: Unnecessary extra details: ----
Note that if <math>z \in X</math> is such that <math>f(z) \neq \infty</math> then either <math>z \in K</math> or else <math>f(z) > 1</math> and <math>\tfrac{1}{f(z)} z \in K</math> so that either way, some positive real scalar multiple of <math>z</math> will belong to <math>K.</math> 
Similarly, if <math>z \in X</math> is such that <math>f(z) \neq 0</math> then some positive real scalar multiple of <math>z</math> will {{em|not}} belong to <math>K.</math> 
It follows that <math>f</math> is identically <math>\,\infty\,</math> (resp. identically <math>0</math>) if and only if <math>K = \varnothing</math> (resp. <math>K = X</math>), which happens if and only if <math>p_K</math> is identically <math>\,\infty\,</math> (resp. identically <math>0</math>). 
So henceforth assume that <math>K</math> is neither empty nor all of <math>X</math> and that <math>f</math> is neither identically <math>\,\infty\,</math> nor identically <math>0.</math> 
--- END ----->

For all real <math>t > 0,</math> <math>f(0) = f(t 0) = t f(0)</math> so by taking <math>t = 2</math> for instance, it follows that either <math>f(0) = 0</math> or <math>f(0) = \infty.</math> 
Let <math>x \in X.</math> 
It remains to show that <math>f(x) = p_K(x).</math> 

It will now be shown that if <math>f(x) = 0</math> or <math>f(x) = \infty</math> then <math>f(x) = p_K(x),</math> so that in particular, it will follow that <math>f(0) = p_K(0).</math> 
So suppose that <math>f(x) = 0</math> or <math>f(x) = \infty;</math> in either case <math>f(t x) = t f(x) = f(x)</math> for all real <math>t > 0.</math> 
Now if <math>f(x) = 0</math> then this implies that that <math>t x \in K</math> for all real <math>t > 0</math> (since <math>f(t x) = 0 \leq 1</math>), which implies that <math>p_K(x) = 0,</math> as desired. 
Similarly, if <math>f(x) = \infty</math> then <math>t x \not\in K</math> for all real <math>t > 0,</math> which implies that <math>p_K(x) = \infty,</math> as desired. 
Thus, it will henceforth be assumed that <math>R := f(x)</math> a positive real number and that <math>x \neq 0</math> (importantly, however, the possibility that <math>p_K(x)</math> is <math>0</math> or <math>\,\infty\,</math> has not yet been ruled out). 

Recall that just like <math>f,</math> the function <math>p_K</math> satisfies <math>p_K(t x) = t p_K(x)</math> for all real <math>t > 0.</math> 
Since <math>0 < \tfrac{1}{R} < \infty,</math> <math>p_K(x)= R = f(x)</math> if and only if <math>p_K\left(\tfrac{1}{R} x\right) = 1 = f\left(\tfrac{1}{R} x\right)</math> so assume without loss of generality that <math>R = 1</math> and it remains to show that <math>p_K\left(\tfrac{1}{R} x\right) = 1.</math> 
Since <math>f(x) = 1,</math> <math>x \in K \subseteq (0, 1] K,</math> which implies that <math>p_K(x) \leq 1</math> (so in particular, <math>p_K(x) \neq \infty</math> is guaranteed). 
It remains to show that <math>p_K(x) \geq 1,</math> which recall happens if and only if <math>x \not\in (0, 1) K.</math> 
So assume for the sake of contradiction that <math>x \in (0, 1) K</math> and let <math>0 < r < 1</math> and <math>k \in K</math> be such that <math>x = r k,</math> where note that <math>k \in K</math> implies that <math>f(k) \leq 1.</math> 
Then <math>1 = f(x) = f(r k) = r f(k) \leq r < 1.</math> <math>\blacksquare</math>
{{collapse bottom}}

This theorem can be extended to characterize certain classes of <math>[- \infty, \infty]</math>-valued maps (for example, real-valued [[sublinear function]]s) in terms of Minkowski functionals. 
For instance, it can be used to describe how every real homogeneous function <math>f : X \to \R</math> (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

===Characterizing Minkowski functionals on star sets===

{{Math theorem|name=Proposition{{sfn|Schechter|1996|pp=313-317}}|math_statement=
Let <math>f : X \to [0, \infty]</math> be any function and <math>K \subseteq X</math> be any subset. 
The following statements are equivalent: 
<ol>
<li><math>f</math> is (strictly) positive homogeneous, <math>f(0) = 0,</math> and 
<math display=block>\{x \in X : f(x) < 1\} \; \subseteq \; K \; \subseteq \; \{x \in X : f(x) \leq 1\}.</math></li>
<li><math>f</math> is the Minkowski functional of <math>K</math> (that is, <math>f = p_K</math>), <math>K</math> contains the origin, and <math>K</math> is [[Star domain|star-shaped]] at the origin.
* The set <math>K</math> is star-shaped at the origin if and only if <math>t k \in K</math> whenever <math>k \in K</math> and <math>0 \leq t \leq 1.</math> A set that is star-shaped at the origin is sometimes called a {{em|star set}}.{{sfn|Schechter|1996|p=303}}</li>
</ol>
}}

===Characterizing Minkowski functionals that are seminorms===

In this next theorem, which follows immediately from the statements above, <math>K</math> is {{em|not}} assumed to be absorbing in <math>X</math> and instead, it is deduced that <math>(0, 1) K</math> is absorbing when <math>p_K</math> is a seminorm. It is also not assumed that <math>K</math> is [[Balanced set|balanced]] (which is a property that <math>K</math> is often required to have); in its place is the weaker condition that <math>(0, 1) s K \subseteq (0, 1) K</math> for all scalars <math>s</math> satisfying <math>|s| = 1.</math> 
The common requirement that <math>K</math> be convex is also weakened to only requiring that <math>(0, 1) K</math> be convex.

{{Math theorem|name=Theorem|math_statement=
Let <math>K</math> be a subset of a real or complex vector space <math>X.</math> 
Then <math>p_K</math> is a [[seminorm]] on <math>X</math> if and only if all of the following conditions hold:
<ol>
<li><math>(0, \infty) K = X</math> (or equivalently, <math>p_K</math> is real-valued).</li>
<li><math>(0, 1) K</math> is convex (or equivalently, <math>p_K</math> is [[subadditivity|subadditive]]).
* It suffices (but is not necessary) for <math>K</math> to be convex.</li>
<li><math>(0, 1) u K \subseteq (0, 1) K</math> for all unit scalars <math>u.</math>
* This condition is satisfied if <math>K</math> is [[Balanced set|balanced]] or more generally if <math>u K \subseteq K</math> for all unit scalars <math>u.</math></li>
</ol>
in which case <math>0 \in K</math> and both <math>(0, 1) K = \{x \in X : p(x) < 1\}</math> and <math>\bigcap_{e > 0} (0, 1 + e) K = \left\{x \in X : p_K(x) \leq 1\right\}</math> will be convex, balanced, and [[Absorbing set|absorbing]] subsets of <math>X.</math> 

Conversely, if <math>f</math> is a seminorm on <math>X</math> then the set <math>V := \{x \in X : f(x) < 1\}</math> satisfies all three of the above conditions (and thus also the conclusions) and also <math>f = p_V;</math> 
moreover, <math>V</math> is necessarily convex, balanced, absorbing, and satisfies <math>(0, 1) V = V = [0, 1] V.</math> 
}}

{{Math theorem|name=Corollary|math_statement=
If <math>K</math> is a convex, balanced, and absorbing subset of a real or complex vector space <math>X,</math> then <math>p_K</math> is a [[seminorm]] on <math>X.</math> 
}}

===Positive sublinear functions and Minkowski functionals===

It may be shown that a real-valued [[subadditive function]] <math>f : X \to \R</math> on an arbitrary [[topological vector space]] <math>X</math> is continuous at the origin if and only if it is uniformly continuous, where if in addition <math>f</math> is nonnegative, then <math>f</math> is continuous if and only if <math>V := \{x \in X : f(x) < 1\}</math> is an open neighborhood in <math>X.</math>{{sfn|Narici|Beckenstein|2011|pp=192-193}} 
If <math>f : X \to \R</math> is subadditive and satisfies <math>f(0) = 0,</math> then <math>f</math> is continuous if and only if its absolute value <math>|f| : X \to [0, \infty)</math> is continuous. 

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

'''Correspondence between open convex sets and positive continuous sublinear functions'''

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=192-193}}|math_statement=
Suppose that <math>X</math> is a [[topological vector space]] (not necessarily [[Locally convex topological vector space|locally convex]] or Hausdorff) over the real or complex numbers. 
Then the non-empty open convex subsets of <math>X</math> are exactly those sets that are of the form <math>z + \{x \in X : p(x) < 1\} = \{x \in X : p(x - z) < 1\}</math> for some <math>z \in X</math> and some positive continuous [[sublinear function]] <math>p</math> on <math>X.</math>
}}

{{collapse top|title=Proof|left=true}}
Let <math>V \neq \varnothing</math> be an open convex subset of <math>X.</math> 
If <math>0 \in V</math> then let <math>z := 0</math> and otherwise let <math>z \in V</math> be arbitrary. 
Let <math>p = p_K : X \to [0, \infty)</math> be the Minkowski functional of <math>K := V - z</math> where this convex open neighborhood of the origin satisfies <math>(0, 1) K = K.</math> 
Then <math>p</math> is a continuous sublinear function on <math>X</math> since <math>V - z</math> is convex, absorbing, and open (however, <math>p</math> is not necessarily a seminorm since it is not necessarily absolutely homogeneous). 
From the properties of Minkowski functionals, we have <math>p_K^{-1}([0, 1)) = (0, 1) K,</math> from which it follows that <math>V - z = \{x \in X : p(x) < 1\}</math> and so 
<math>V = z + \{x \in X : p(x) < 1\}.</math> 
Since <math>z + \{x \in X : p(x) < 1\} = \{x \in X : p(x - z) < 1\},</math> this completes the proof. <math>\blacksquare</math>
{{collapse bottom}}

==See also==

* {{annotated link|Asymmetric norm}}
* {{annotated link|Auxiliary normed space}}
* {{annotated link|Cauchy's functional equation}}
* {{annotated link|Finest locally convex topology}}
* {{annotated link|Finsler manifold}}
* {{annotated link|Hadwiger's theorem}}
* {{annotated link|Hugo Hadwiger}}
* {{annotated link|Locally convex topological vector space}}
* {{annotated link|Morphological image processing}}
* {{annotated link|Norm (mathematics)}}
* {{annotated link|Seminorm}}
* {{annotated link|Topological vector space}}

==Notes==

{{reflist|group=note}}
{{reflist|group=proof}}

==References==

{{reflist}}

* {{Berberian Lectures in Functional Analysis and Operator Theory}} <!--{{sfn|Berberian|2014|p=}}-->
* {{Bourbaki Topological Vector Spaces}} <!--{{sfn|Bourbaki|1987|p=}}-->
* {{Conway A Course in Functional Analysis}} <!--{{sfn|Conway|1990|p=}}-->
* {{Diestel The Metric Theory of Tensor Products Grothendieck's Résumé Revisited}} <!--{{sfn|Diestel|2008|p=}}-->
* {{Dineen Complex Analysis in Locally Convex Spaces}} <!--{{sfn|Dineen|1981|p=}}-->
* {{Dunford Schwartz Linear Operators Part 1 General Theory}} <!--{{sfn|Dunford|1988|p=}}-->
* {{Edwards Functional Analysis Theory and Applications}} <!--{{sfn|Edwards|1995|p=}}-->
* {{Grothendieck Topological Vector Spaces}} <!--{{sfn|Grothendieck|1973|p=}}-->
* {{Hogbe-Nlend Bornologies and Functional Analysis}} <!--{{sfn|Hogbe-Nlend|1977|p=}}-->
* {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}} <!--{{sfn|Hogbe-Nlend|1981|p=}}-->
* {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}} <!--{{sfn|Husain|1978|p=}}-->
* {{Keller Differential Calculus in Locally Convex Spaces}} <!--{{sfn|Keller|1974|p=}}-->
* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|p=}}-->
* {{Kubrusly The Elements of Operator Theory 2nd Edition 2011}} <!--{{sfn|Kubrusly|2011|p=}}-->
* {{Jarchow Locally Convex Spaces}} <!--{{sfn|Jarchow|1981|p=}}-->
* {{Köthe Topological Vector Spaces I}} <!--{{sfn|Köthe|1969|p=}}-->
* {{Köthe Topological Vector Spaces II}} <!--{{sfn|Köthe|1979|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Pietsch Nuclear Locally Convex Spaces|edition=2}} <!--{{sfn|Pietsch|1979|p=}}-->
* {{Robertson Topological Vector Spaces}} <!--{{sfn|Robertson|1980|p=}}-->
* {{Rudin Walter Functional Analysis|edition=2}} <!--{{sfn|Rudin|1991|p=}}-->
* {{cite book|last=Thompson|first=Anthony C.|title=Minkowski Geometry|url=https://archive.org/details/minkowskigeometr0000thom|url-access=registration|series=Encyclopedia of Mathematics and Its Applications |publisher=[[Cambridge University Press]]|year=1996|isbn=0-521-40472-X }}
* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|1999|p=}}-->
* {{Schechter Handbook of Analysis and Its Foundations}} <!--{{sfn|Schechter|1996|p=}}-->
* {{cite book|last=Schaefer|first=H. H.|title=Topological Vector Spaces|publisher=Springer New York Imprint Springer|publication-place=New York, NY|year=1999|isbn=978-1-4612-7155-0|oclc=840278135}} <!--{{sfn|Schaefer|1999|p=}}-->
* {{Swartz An Introduction to Functional Analysis}} <!--{{sfn|Swartz|1992|p=}}-->
* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!--{{sfn|Wilansky|2013|p=}}-->
* {{Wong Schwartz Spaces, Nuclear Spaces, and Tensor Products}} <!--{{sfn|Wong|1979|p=}}-->

==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). https://doi.org/10.3390/en13225976

{{Functional analysis}}
{{Topological vector spaces}}
{{Convex analysis and variational analysis}}

[[Category:Convex analysis]]
[[Category:Functional analysis]]
[[Category:Hermann Minkowski]]