# 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{\textstyle K} is a subset of a real or complex vector space X, {\textstyle X, } then the Minkowski functional or gauge of K{\textstyle K} is defined to be the function p$_K$:X→[0, ∞], {\textstyle p_{K}:X\to [0, \infty ], } valued in the extended real numbers, defined by $p_{K}(x):=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}\quad {\text{ for every }}x\in X,$ where the infimum of the empty set is defined to be positive infinity ∞{\textstyle \, \infty \, } (which is not a real number so that p$_K$(x){\textstyle p_{K}(x)} would then not be real-valued).
The set K{\textstyle K} is often assumed/picked to have properties, such as being an absorbing disk in X{\textstyle X}, that guarantee that p$_K${\textstyle p_{K}} will be a real-valued seminorm on X.{\textstyle X.} In fact, every seminorm p{\textstyle p} on X{\textstyle X} is equal to the Minkowski functional (that is, p=p$_K${\textstyle p=p_{K}}) of any subset K{\textstyle K} of X{\textstyle X} satisfying
$\{x\in X:p(x)<1\}\subseteq K\subseteq \{x\in X:p(x)\leq 1\}$
(where all three of these sets are necessarily absorbing in X{\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 X{\textstyle X} into certain algebraic properties of a function on X.{\textstyle X.}
The Minkowski function is always non-negative (meaning p$_K$≥0{\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, p$_K${\textstyle p_{K}} might not be real-valued since for any given x∈X, {\textstyle x\in X, } the value p$_K$(x){\textstyle p_{K}(x)} is a real number if and only if {r>0:x∈rK}{\textstyle \{r>0:x\in rK\}} is not empty. Consequently, K{\textstyle K} is usually assumed to have properties (such as being absorbing in X, {\textstyle X, } for instance) that will guarantee that p$_K${\textstyle p_{K}} is real-valued.

## Definition

Let K{\textstyle K} be a subset of a real or complex vector space X.{\textstyle X.} Define the gauge of K{\textstyle K} or the Minkowski functional associated with or induced by K{\textstyle K} as being the function p$_K$:X→[0, ∞], {\textstyle p_{K}:X\to [0, \infty ], } valued in the extended real numbers, defined by
$p_{K}(x):=\inf\{r>0:x\in rK\},$
(recall that the infimum of the empty set is ∞{\textstyle \, \infty }, that is, inf∅=∞{\textstyle \inf \varnothing =\infty }). Here, {r>0:x∈rK}{\textstyle \{r>0:x\in rK\}} is shorthand for {r∈R:r>0 and x∈rK}.{\textstyle \{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}.}
For any x∈X, {\textstyle x\in X, } p$_K$(x)≠∞{\textstyle p_{K}(x)\neq \infty } if and only if {r>0:x∈rK}{\textstyle \{r>0:x\in rK\}} is not empty. The arithmetic operations on R{\textstyle \mathbb {R} } can be extended to operate on ±∞, {\textstyle \pm \infty , } where r/±∞ :=0{\textstyle {\frac {r}{\pm \infty }}:=0} for all non-zero real −∞<r<∞.{\textstyle -\infty <r<\infty .} The products 0⋅∞{\textstyle 0\cdot \infty } and 0⋅−∞{\textstyle 0\cdot -\infty } remain undefined.

### Some conditions making a gauge real-valued

In the field of convex analysis, the map p$_K${\textstyle p_{K}} taking on the value of ∞{\textstyle \, \infty \, } is not necessarily an issue. However, in functional analysis p$_K${\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 {r>0:x∈rK}{\textstyle \{r>0:x\in rK\}} is non-empty for every x∈X.{\textstyle x\in X.}
In order for p$_K${\textstyle p_{K}} to be real-valued, it suffices for the origin of X{\textstyle X} to belong to the algebraic interior or core of K{\textstyle K} in X.{\textstyle X.} If K{\textstyle K} is absorbing in X, {\textstyle X, } where recall that this implies that 0∈K, {\textstyle 0\in K, } then the origin belongs to the algebraic interior of K{\textstyle K} in X{\textstyle X} and thus p$_K${\textstyle p_{K}} is real-valued. Characterizations of when p$_K${\textstyle p_{K}} is real-valued are given below.

## Motivating examples
### Example 1

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

### Example 2

Let X{\textstyle X} be a vector space without topology with underlying scalar field K.{\textstyle \mathbb {K} .} Let f:X→K{\textstyle f:X\to \mathbb {K} } be any linear functional on X{\textstyle X} (not necessarily continuous). Fix a>0.{\textstyle a>0.} Let K{\textstyle K} be the set $K:=\{x\in X:|f(x)|\leq a\}$ and let p$_K${\textstyle p_{K}} be the Minkowski functional of K.{\textstyle K.} Then $p_{K}(x)={\frac {1}{a}}|f(x)|\quad {\text{ for all }}x\in X.$ The function p$_K${\textstyle p_{K}} has the following properties:
* It is subadditive: p$_K$(x+y)≤p$_K$(x)+p$_K$(y).{\textstyle p_{K}(x+y)\leq p_{K}(x)+p_{K}(y).}
* It is absolutely homogeneous: p$_K$(sx)=|s|p$_K$(x){\textstyle p_{K}(sx)=|s|p_{K}(x)} for all scalars s.{\textstyle s.}
* It is nonnegative: p$_K$≥0.{\textstyle p_{K}\geq 0.}
Therefore, p$_K${\textstyle p_{K}} is a seminorm on X, {\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, p$_K$(x)=0{\textstyle p_{K}(x)=0} need not imply x=0.{\textstyle x=0.} In the above example, one can take a nonzero x{\textstyle x} from the kernel of f.{\textstyle f.} Consequently, the resulting topology need not be Hausdorff.

## Common conditions guaranteeing gauges are seminorms

To guarantee that p$_K$(0)=0, {\textstyle p_{K}(0)=0, } it will henceforth be assumed that 0∈K.{\textstyle 0\in K.}
In order for p$_K${\textstyle p_{K}} to be a seminorm, it suffices for K{\textstyle K} to be a disk (that is, convex and balanced) and absorbing in X, {\textstyle X, } which are the most common assumption placed on K.{\textstyle K.}
Theorem — If K{\textstyle K} is an absorbing disk in a vector space X{\textstyle X} then the Minkowski functional of K, {\textstyle K, } which is the map p$_K$:X→[0, ∞){\textstyle p_{K}:X\to [0, \infty )} defined by $p_{K}(x):=\inf\{r>0:x\in rK\},$ is a seminorm on X.{\textstyle X.} Moreover, $p_{K}(x)={\frac {1}{\sup\{r>0:rx\in K\}}}.$
More generally, if K{\textstyle K} is convex and the origin belongs to the algebraic interior of K, {\textstyle K, } then p$_K${\textstyle p_{K}} is a nonnegative sublinear functional on X, {\textstyle X, } which implies in particular that it is subadditive and positive homogeneous. If K{\textstyle K} is absorbing in X{\textstyle X} then p$_{[0, 1]K}${\textstyle p_{[0, 1]K}} is positive homogeneous, meaning that p$_{[0, 1]K}$(sx)=sp$_{[0, 1]K}$(x){\textstyle p_{[0, 1]K}(sx)=sp_{[0, 1]K}(x)} for all real s≥0, {\textstyle s\geq 0, } where [0, 1]K={tk:t∈[0, 1], k∈K}.{\textstyle [0, 1]K=\{tk:t\in [0, 1], k\in K\}.} If q{\textstyle q} is a nonnegative real-valued function on X{\textstyle X} that is positive homogeneous, then the sets U:={x∈X:q(x)<1}{\textstyle U:=\{x\in X:q(x)<1\}} and D:={x∈X:q(x)≤1}{\textstyle D:=\{x\in X:q(x)\leq 1\}} satisfy [0, 1]U=U{\textstyle [0, 1]U=U} and [0, 1]D=D;{\textstyle [0, 1]D=D;} if in addition q{\textstyle q} is absolutely homogeneous then both U{\textstyle U} and D{\textstyle D} are balanced.

### Gauges of absorbing disks

Arguably the most common requirements placed on a set K{\textstyle K} to guarantee that p$_K${\textstyle p_{K}} is a seminorm are that K{\textstyle K} be an absorbing disk in X.{\textstyle X.} Due to how common these assumptions are, the properties of a Minkowski functional p$_K${\textstyle p_{K}} when K{\textstyle K} is an absorbing disk will now be investigated. Since all of the results mentioned above made few (if any) assumptions on K, {\textstyle K, } they can be applied in this special case.
Theorem — Assume that K{\textstyle K} is an absorbing subset of X.{\textstyle X.} It is shown that:
* If K{\textstyle K} is convex then p$_K${\textstyle p_{K}} is subadditive.
* If K{\textstyle K} is balanced then p$_K${\textstyle p_{K}} is absolutely homogeneous; that is, p$_K$(sx)=|s|p$_K$(x){\textstyle p_{K}(sx)=|s|p_{K}(x)} for all scalars s.{\textstyle s.}

### Algebraic properties

Let X{\textstyle X} be a real or complex vector space and let K{\textstyle K} be an absorbing disk in X.{\textstyle X.}
* p$_K${\textstyle p_{K}} is a seminorm on X.{\textstyle X.}
* p$_K${\textstyle p_{K}} is a norm on X{\textstyle X} if and only if K{\textstyle K} does not contain a non-trivial vector subspace.
* p$_{sK}$=1/|s| p$_K${\textstyle p_{sK}={\frac {1}{|s|}}p_{K}} for any scalar s≠0.{\textstyle s\neq 0.}
* If J{\textstyle J} is an absorbing disk in X{\textstyle X} and J⊆K{\textstyle J\subseteq K} then p$_K$≤p$_J$.{\textstyle p_{K}\leq p_{J}.}
* If K{\textstyle K} is a set satisfying {x∈X:p(x)<1}⊆K⊆{x∈X:p(x)≤1}{\textstyle \{x\in X:p(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p(x)\leq 1\}} then K{\textstyle K} is absorbing in X{\textstyle X} and p=p$_K$, {\textstyle p=p_{K}, } where p$_K${\textstyle p_{K}} is the Minkowski functional associated with K;{\textstyle K;} that is, it is the gauge of K.{\textstyle K.}
* In particular, if K{\textstyle K} is as above and q{\textstyle q} is any seminorm on X, {\textstyle X, } then q=p{\textstyle q=p} if and only if {x∈X:q(x)<1}⊆K⊆{x∈X:q(x)≤1}.{\textstyle \{x\in X:q(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:q(x)\leq 1\}.}
* If x∈X{\textstyle x\in X} satisfies p$_K$(x)<1{\textstyle p_{K}(x)<1} then x∈K.{\textstyle x\in K.}

### Topological properties

Assume that X{\textstyle X} is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let K{\textstyle K} be an absorbing disk in X.{\textstyle X.} Then
$\operatorname {Int} _{X}K\;\subseteq \;\{x\in X:p_{K}(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:p_{K}(x)\leq 1\}\;\subseteq \;\operatorname {Cl} _{X}K,$
where Int$_X$⁡K{\textstyle \operatorname {Int} _{X}K} is the topological interior and Cl$_X$⁡K{\textstyle \operatorname {Cl} _{X}K} is the topological closure of K{\textstyle K} in X.{\textstyle X.} Importantly, it was not assumed that p$_K${\textstyle p_{K}} was continuous nor was it assumed that K{\textstyle K} had any topological properties.
Moreover, the Minkowski functional p$_K${\textstyle p_{K}} is continuous if and only if K{\textstyle K} is a neighborhood of the origin in X.{\textstyle X.} If p$_K${\textstyle p_{K}} is continuous then $\operatorname {Int} _{X}K=\{x\in X:p_{K}(x)<1\}\quad {\text{ and }}\quad \operatorname {Cl} _{X}K=\{x\in X:p_{K}(x)\leq 1\}.$

## Minimal requirements on the set

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

### Properties

All results in this section may be applied to the case where K{\textstyle K} is an absorbing disk.
Throughout, K{\textstyle K} is any subset of X.{\textstyle X.}
Summary — Suppose that K{\textstyle K} is a subset of a real or complex vector space X.{\textstyle X.}
* Strict positive homogeneity: p$_K$(rx)=rp$_K$(x){\textstyle p_{K}(rx)=rp_{K}(x)} for all x∈X{\textstyle x\in X} and all positive real r>0.{\textstyle r>0.}
** Positive/Nonnegative homogeneity: p$_K${\textstyle p_{K}} is nonnegative homogeneous if and only if p$_K${\textstyle p_{K}} is real-valued.
*** A map p{\textstyle p} is called nonnegative homogeneous if p(rx)=rp(x){\textstyle p(rx)=rp(x)} for all x∈X{\textstyle x\in X} and all nonnegative real r≥0.{\textstyle r\geq 0.} Since 0⋅∞{\textstyle 0\cdot \infty } is undefined, a map that takes infinity as a value is not nonnegative homogeneous.
* Real-values: (0, ∞)K{\textstyle (0, \infty )K} is the set of all points on which p$_K${\textstyle p_{K}} is real valued. So p$_K${\textstyle p_{K}} is real-valued if and only if (0, ∞)K=X, {\textstyle (0, \infty )K=X, } in which case 0∈K.{\textstyle 0\in K.}
** Value at 0{\textstyle 0}: p$_K$(0)≠∞{\textstyle p_{K}(0)\neq \infty } if and only if 0∈K{\textstyle 0\in K} if and only if p$_K$(0)=0.{\textstyle p_{K}(0)=0.}
** Null space: If x∈X{\textstyle x\in X} then p$_K$(x)=0{\textstyle p_{K}(x)=0} if and only if (0, ∞)x⊆(0, 1)K{\textstyle (0, \infty )x\subseteq (0, 1)K} if and only if there exists a divergent sequence of positive real numbers t$_1$, t$_2$, t$_3$, ⋯→∞{\textstyle t_{1}, t_{2}, t_{3}, \cdots \to \infty } such that t$_n$x∈K{\textstyle t_{n}x\in K} for all n.{\textstyle n.} Moreover, the zero set of p$_K${\textstyle p_{K}} is ker⁡p$_K$ =$^{def}$ {y∈X:p$_K$(y)=0}=⋂$_{e>0}$(0, e)K.{\textstyle \ker p_{K}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{y\in X:p_{K}(y)=0\right\}={\textstyle \bigcap \limits _{e>0}}(0, e)K.}
* Comparison to a constant: If 0≤r≤∞{\textstyle 0\leq r\leq \infty } then for any x∈X, {\textstyle x\in X, } p$_K$(x)<r{\textstyle p_{K}(x)<r} if and only if x∈(0, r)K;{\textstyle x\in (0, r)K;} this can be restated as: If 0≤r≤∞{\textstyle 0\leq r\leq \infty } then p$_K$$^{−1}$([0, r))=(0, r)K.{\textstyle p_{K}^{-1}([0, r))=(0, r)K.}
** It follows that if 0≤R<∞{\textstyle 0\leq R<\infty } is real then p$_K$$^{−1}$([0, R])=⋂$_{e>0}$(0, R+e)K, {\textstyle p_{K}^{-1}([0, R])={\textstyle \bigcap \limits _{e>0}}(0, R+e)K, } where the set on the right hand side denotes ⋂$_{e>0}$[(0, R+e)K]{\textstyle {\textstyle \bigcap \limits _{e>0}}[(0, R+e)K]} and not its subset [⋂$_{e>0}$(0, R+e)]K=(0, R]K.{\textstyle \left[{\textstyle \bigcap \limits _{e>0}}(0, R+e)\right]K=(0, R]K.} If R>0{\textstyle R>0} then these sets are equal if and only if K{\textstyle K} contains {y∈X:p$_K$(y)=1}.{\textstyle \left\{y\in X:p_{K}(y)=1\right\}.}
** In particular, if x∈RK{\textstyle x\in RK} or x∈(0, R]K{\textstyle x\in (0, R]K} then p$_K$(x)≤R, {\textstyle p_{K}(x)\leq R, } but importantly, the converse is not necessarily true.
* Gauge comparison: For any subset L⊆X, {\textstyle L\subseteq X, } p$_K$≤p$_L${\textstyle p_{K}\leq p_{L}} if and only if (0, 1)L⊆(0, 1)K;{\textstyle (0, 1)L\subseteq (0, 1)K;} thus p$_L$=p$_K${\textstyle p_{L}=p_{K}} if and only if (0, 1)L=(0, 1)K.{\textstyle (0, 1)L=(0, 1)K.}
** The assignment L↦p$_L${\textstyle L\mapsto p_{L}} is order-reversing in the sense that if K⊆L{\textstyle K\subseteq L} then p$_L$≤p$_K$.{\textstyle p_{L}\leq p_{K}.}
** Because the set L:=(0, 1)K{\textstyle L:=(0, 1)K} satisfies (0, 1)L=(0, 1)K, {\textstyle (0, 1)L=(0, 1)K, } it follows that replacing K{\textstyle K} with p$_K$$^{−1}$([0, 1))=(0, 1)K{\textstyle p_{K}^{-1}([0, 1))=(0, 1)K} will not change the resulting Minkowski functional. The same is true of L:=(0, 1]K{\textstyle L:=(0, 1]K} and of L:=p$_K$$^{−1}$([0, 1]).{\textstyle L:=p_{K}^{-1}([0, 1]).}
** If D =$^{def}$ {y∈X:p$_K$(y)=1 or p$_K$(y)=0}{\textstyle D~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{y\in X:p_{K}(y)=1{\text{ or }}p_{K}(y)=0\right\}} then p$_D$=p$_K${\textstyle p_{D}=p_{K}} and D{\textstyle D} has the particularly nice property that if r>0{\textstyle r>0} is real then x∈rD{\textstyle x\in rD} if and only if p$_D$(x)=r{\textstyle p_{D}(x)=r} or p$_D$(x)=0.{\textstyle p_{D}(x)=0.} Moreover, if r>0{\textstyle r>0} is real then p$_D$(x)≤r{\textstyle p_{D}(x)\leq r} if and only if x∈(0, r]D.{\textstyle x\in (0, r]D.}
* Subadditive/Triangle inequality: p$_K${\textstyle p_{K}} is subadditive if and only if (0, 1)K{\textstyle (0, 1)K} is convex. If K{\textstyle K} is convex then so are both (0, 1)K{\textstyle (0, 1)K} and (0, 1]K{\textstyle (0, 1]K} and moreover, p$_K${\textstyle p_{K}} is subadditive.
* Scaling the set: If s≠0{\textstyle s\neq 0} is a scalar then p$_{sK}$(y)=p$_K$( 1/s y){\textstyle p_{sK}(y)=p_{K}\left({\tfrac {1}{s}}y\right)} for all y∈X.{\textstyle y\in X.} Thus if 0<r<∞{\textstyle 0<r<\infty } is real then p$_{rK}$(y)=p$_K$( 1/r y)=1/r p$_K$(y).{\textstyle p_{rK}(y)=p_{K}\left({\tfrac {1}{r}}y\right)={\tfrac {1}{r}}p_{K}(y).}
* Symmetric: p$_K${\textstyle p_{K}} is symmetric (meaning that p$_K$(−y)=p$_K$(y){\textstyle p_{K}(-y)=p_{K}(y)} for all y∈X{\textstyle y\in X}) if and only if (0, 1)K{\textstyle (0, 1)K} is a symmetric set (meaning that (0, 1)K=−(0, 1)K{\textstyle (0, 1)K=-(0, 1)K}), which happens if and only if p$_K$=p$_{−K}$.{\textstyle p_{K}=p_{-K}.}
* Absolute homogeneity: p$_K$(ux)=p$_K$(x){\textstyle p_{K}(ux)=p_{K}(x)} for all x∈X{\textstyle x\in X} and all unit length scalars u{\textstyle u} if and only if (0, 1)uK⊆(0, 1)K{\textstyle (0, 1)uK\subseteq (0, 1)K} for all unit length scalars u, {\textstyle u, } in which case p$_K$(sx)=|s|p$_K$(x){\textstyle p_{K}(sx)=|s|p_{K}(x)} for all x∈X{\textstyle x\in X} and all non-zero scalars s≠0.{\textstyle s\neq 0.} If in addition p$_K${\textstyle p_{K}} is also real-valued then this holds for all scalars s{\textstyle s} (that is, p$_K${\textstyle p_{K}} is absolutely homogeneous).
** (0, 1)uK⊆(0, 1)K{\textstyle (0, 1)uK\subseteq (0, 1)K} for all unit length u{\textstyle u} if and only if (0, 1)uK=(0, 1)K{\textstyle (0, 1)uK=(0, 1)K} for all unit length u.{\textstyle u.}
** sK⊆K{\textstyle sK\subseteq K} for all unit scalars s{\textstyle s} if and only if sK=K{\textstyle sK=K} for all unit scalars s;{\textstyle s;} if this is the case then (0, 1)K=(0, 1)sK{\textstyle (0, 1)K=(0, 1)sK} for all unit scalars s.{\textstyle s.}
** The Minkowski functional of any balanced set is a balanced function.
* Absorbing: If K{\textstyle K} is convex or balanced and if (0, ∞)K=X{\textstyle (0, \infty )K=X} then K{\textstyle K} is absorbing in X.{\textstyle X.}
** If a set A{\textstyle A} is absorbing in X{\textstyle X} and A⊆K{\textstyle A\subseteq K} then K{\textstyle K} is absorbing in X.{\textstyle X.}
** If K{\textstyle K} is convex and 0∈K{\textstyle 0\in K} then [0, 1]K=K, {\textstyle [0, 1]K=K, } in which case (0, 1)K⊆K.{\textstyle (0, 1)K\subseteq K.}
* Restriction to a vector subspace: If S{\textstyle S} is a vector subspace of X{\textstyle X} and if p$_{K∩S}$:S→[0, ∞]{\textstyle p_{K\cap S}:S\to [0, \infty ]} denotes the Minkowski functional of K∩S{\textstyle K\cap S} on S, {\textstyle S, } then p$_K$|$_S$=p$_{K∩S}$, {\textstyle p_{K}{\big \vert }_{S}=p_{K\cap S}, } where p$_K$|$_S${\textstyle p_{K}{\big \vert }_{S}} denotes the restriction of p$_K${\textstyle p_{K}} to S.{\textstyle S.}

### Examples

* If L{\textstyle {\mathcal {L}}} is a non-empty collection of subsets of X{\textstyle X} then p$_{∪L}$(x)=inf{p$_L$(x):L∈L}{\textstyle p_{\cup {\mathcal {L}}}(x)=\inf \left\{p_{L}(x):L\in {\mathcal {L}}\right\}} for all x∈X, {\textstyle x\in X, } where ∪L =$^{def}$ ⋃$_{L∈L}$L.{\textstyle \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)}{\textstyle p_{K\cup L}(x)=\min \left\{p_{K}(x), p_{L}(x)\right\}} for all x∈X.{\textstyle x\in X.}
* If L{\textstyle {\mathcal {L}}} is a non-empty collection of subsets of X{\textstyle X} and I⊆X{\textstyle I\subseteq X} satisfies
$\left\{x\in X:p_{L}(x)<1{\text{ for all }}L\in {\mathcal {L}}\right\}\quad \subseteq \quad I\quad \subseteq \quad \left\{x\in X:p_{L}(x)\leq 1{\text{ for all }}L\in {\mathcal {L}}\right\}$ then p$_I$(x)=sup{p$_L$(x):L∈L}{\textstyle p_{I}(x)=\sup \left\{p_{L}(x):L\in {\mathcal {L}}\right\}} for all x∈X.{\textstyle x\in X.}
The following examples show that the containment (0, R]K⊆⋂$_{e>0}$(0, R+e)K{\textstyle (0, R]K\;\subseteq \;{\textstyle \bigcap \limits _{e>0}}(0, R+e)K} could be proper.
Example: If R=0{\textstyle R=0} and K=X{\textstyle K=X} then (0, R]K=(0, 0]X=∅X=∅{\textstyle (0, R]K=(0, 0]X=\varnothing X=\varnothing } but ⋂$_{e>0}$(0, e)K=⋂$_{e>0}$X=X, {\textstyle {\textstyle \bigcap \limits _{e>0}}(0, e)K={\textstyle \bigcap \limits _{e>0}}X=X, } which shows that its possible for (0, R]K{\textstyle (0, R]K} to be a proper subset of ⋂$_{e>0}$(0, R+e)K{\textstyle {\textstyle \bigcap \limits _{e>0}}(0, R+e)K} when R=0.{\textstyle R=0.} ◼{\textstyle \blacksquare }
The next example shows that the containment can be proper when R=1;{\textstyle R=1;} the example may be generalized to any real R>0.{\textstyle R>0.} Assuming that [0, 1]K⊆K, {\textstyle [0, 1]K\subseteq K, } the following example is representative of how it happens that x∈X{\textstyle x\in X} satisfies p$_K$(x)=1{\textstyle p_{K}(x)=1} but x∉(0, 1]K.{\textstyle x\not \in (0, 1]K.}
Example: Let x∈X{\textstyle x\in X} be non-zero and let K=[0, 1)x{\textstyle K=[0, 1)x} so that [0, 1]K=K{\textstyle [0, 1]K=K} and x∉K.{\textstyle x\not \in K.} From x∉(0, 1)K=K{\textstyle x\not \in (0, 1)K=K} it follows that p$_K$(x)≥1.{\textstyle p_{K}(x)\geq 1.} That p$_K$(x)≤1{\textstyle p_{K}(x)\leq 1} follows from observing that for every e>0, {\textstyle e>0, } (0, 1+e)K=[0, 1+e)([0, 1)x)=[0, 1+e)x, {\textstyle (0, 1+e)K=[0, 1+e)([0, 1)x)=[0, 1+e)x, } which contains x.{\textstyle x.} Thus p$_K$(x)=1{\textstyle p_{K}(x)=1} and x∈⋂$_{e>0}$(0, 1+e)K.{\textstyle x\in {\textstyle \bigcap \limits _{e>0}}(0, 1+e)K.} However, (0, 1]K=(0, 1]([0, 1)x)=[0, 1)x=K{\textstyle (0, 1]K=(0, 1]([0, 1)x)=[0, 1)x=K} so that x∉(0, 1]K, {\textstyle x\not \in (0, 1]K, } as desired. ◼{\textstyle \blacksquare }

### Positive homogeneity characterizes Minkowski functionals

The next theorem shows that Minkowski functionals are exactly those functions f:X→[0, ∞]{\textstyle f:X\to [0, \infty ]} that have a certain purely algebraic property that is commonly encountered.
Theorem — Let f:X→[0, ∞]{\textstyle f:X\to [0, \infty ]} be any function. The following statements are equivalent:
* Strict positive homogeneity: f(tx)=tf(x){\textstyle \;f(tx)=tf(x)} for all x∈X{\textstyle x\in X} and all positive real t>0.{\textstyle t>0.}
** This statement is equivalent to: f(tx)≤tf(x){\textstyle f(tx)\leq tf(x)} for all x∈X{\textstyle x\in X} and all positive real t>0.{\textstyle t>0.}
* f{\textstyle f} is a Minkowski functional: meaning that there exists a subset S⊆X{\textstyle S\subseteq X} such that f=p$_S$.{\textstyle f=p_{S}.}
* f=p$_K${\textstyle f=p_{K}} where K:={x∈X:f(x)≤1}.{\textstyle K:=\{x\in X:f(x)\leq 1\}.}
* f=p$_V${\textstyle f=p_{V}\, } where V:={x∈X:f(x)<1}.{\textstyle V\, :=\{x\in X:f(x)<1\}.}
Moreover, if f{\textstyle f} never takes on the value ∞{\textstyle \, \infty \, } (so that the product 0⋅f(x){\textstyle 0\cdot f(x)} is always well-defined) then this list may be extended to include:
* Positive/Nonnegative homogeneity: f(tx)=tf(x){\textstyle f(tx)=tf(x)} for all x∈X{\textstyle x\in X} and all nonnegative real t≥0{\textstyle t\geq 0}.
This theorem can be extended to characterize certain classes of [−∞, ∞]{\textstyle [-\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{\textstyle 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

Proposition — Let f:X→[0, ∞]{\textstyle f:X\to [0, \infty ]} be any function and K⊆X{\textstyle K\subseteq X} be any subset. The following statements are equivalent:
* f{\textstyle f} is (strictly) positive homogeneous, f(0)=0, {\textstyle f(0)=0, } and $\{x\in X:f(x)<1\}\;\subseteq \;K\;\subseteq \;\{x\in X:f(x)\leq 1\}.$
* f{\textstyle f} is the Minkowski functional of K{\textstyle K} (that is, f=p$_K${\textstyle f=p_{K}}), K{\textstyle K} contains the origin, and K{\textstyle K} is star-shaped at the origin.
** The set K{\textstyle K} is star-shaped at the origin if and only if tk∈K{\textstyle tk\in K} whenever k∈K{\textstyle k\in K} and 0≤t≤1.{\textstyle 0\leq t\leq 1.} A set that is star-shaped at the origin is sometimes called a star set.

### Characterizing Minkowski functionals that are seminorms

In this next theorem, which follows immediately from the statements above, K{\textstyle K} is not assumed to be absorbing in X{\textstyle X} and instead, it is deduced that (0, 1)K{\textstyle (0, 1)K} is absorbing when p$_K${\textstyle p_{K}} is a seminorm. It is also not assumed that K{\textstyle K} is balanced (which is a property that K{\textstyle K} is often required to have); in its place is the weaker condition that (0, 1)sK⊆(0, 1)K{\textstyle (0, 1)sK\subseteq (0, 1)K} for all scalars s{\textstyle s} satisfying |s|=1.{\textstyle |s|=1.} The common requirement that K{\textstyle K} be convex is also weakened to only requiring that (0, 1)K{\textstyle (0, 1)K} be convex.
Theorem — Let K{\textstyle K} be a subset of a real or complex vector space X.{\textstyle X.} Then p$_K${\textstyle p_{K}} is a seminorm on X{\textstyle X} if and only if all of the following conditions hold:
* (0, ∞)K=X{\textstyle (0, \infty )K=X} (or equivalently, p$_K${\textstyle p_{K}} is real-valued).
* (0, 1)K{\textstyle (0, 1)K} is convex (or equivalently, p$_K${\textstyle p_{K}} is subadditive).
** It suffices (but is not necessary) for K{\textstyle K} to be convex.
* (0, 1)uK⊆(0, 1)K{\textstyle (0, 1)uK\subseteq (0, 1)K} for all unit scalars u.{\textstyle u.}
** This condition is satisfied if K{\textstyle K} is balanced or more generally if uK⊆K{\textstyle uK\subseteq K} for all unit scalars u.{\textstyle u.}
in which case 0∈K{\textstyle 0\in K} and both (0, 1)K={x∈X:p(x)<1}{\textstyle (0, 1)K=\{x\in X:p(x)<1\}} and ⋂$_{e>0}$(0, 1+e)K={x∈X:p$_K$(x)≤1}{\textstyle \bigcap _{e>0}(0, 1+e)K=\left\{x\in X:p_{K}(x)\leq 1\right\}} will be convex, balanced, and absorbing subsets of X.{\textstyle X.}
Conversely, if f{\textstyle f} is a seminorm on X{\textstyle X} then the set V:={x∈X:f(x)<1}{\textstyle V:=\{x\in X:f(x)<1\}} satisfies all three of the above conditions (and thus also the conclusions) and also f=p$_V$;{\textstyle f=p_{V};} moreover, V{\textstyle V} is necessarily convex, balanced, absorbing, and satisfies (0, 1)V=V=[0, 1]V.{\textstyle (0, 1)V=V=[0, 1]V.}
Corollary — If K{\textstyle K} is a convex, balanced, and absorbing subset of a real or complex vector space X, {\textstyle X, } then p$_K${\textstyle p_{K}} is a seminorm on X.{\textstyle X.}

### Positive sublinear functions and Minkowski functionals

It may be shown that a real-valued subadditive function f:X→R{\textstyle f:X\to \mathbb {R} } on an arbitrary topological vector space X{\textstyle X} is continuous at the origin if and only if it is uniformly continuous, where if in addition f{\textstyle f} is nonnegative, then f{\textstyle f} is continuous if and only if V:={x∈X:f(x)<1}{\textstyle V:=\{x\in X:f(x)<1\}} is an open neighborhood in X.{\textstyle X.} If f:X→R{\textstyle f:X\to \mathbb {R} } is subadditive and satisfies f(0)=0, {\textstyle f(0)=0, } then f{\textstyle f} is continuous if and only if its absolute value |f|:X→[0, ∞){\textstyle |f|:X\to [0, \infty )} is continuous.
A nonnegative sublinear function is a nonnegative homogeneous function f:X→[0, ∞){\textstyle f:X\to [0, \infty )} that satisfies the triangle inequality. It follows immediately from the results below that for such a function f, {\textstyle f, } if V:={x∈X:f(x)<1}{\textstyle V:=\{x\in X:f(x)<1\}} then f=p$_V$.{\textstyle f=p_{V}.} Given K⊆X, {\textstyle K\subseteq X, } the Minkowski functional p$_K${\textstyle 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{\textstyle (0, \infty )K=X} and (0, 1)K{\textstyle (0, 1)K} is convex.

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

Theorem — Suppose that X{\textstyle X} is a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the non-empty open convex subsets of X{\textstyle X} are exactly those sets that are of the form z+{x∈X:p(x)<1}={x∈X:p(x−z)<1}{\textstyle z+\{x\in X:p(x)<1\}=\{x\in X:p(x-z)<1\}} for some z∈X{\textstyle z\in X} and some positive continuous sublinear function p{\textstyle p} on X.{\textstyle X.}