Minkowski functional

In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space.

If ${\textstyle K}$ is a subset of a real or complex vector space ${\textstyle X,}$ then the Minkowski functional or gauge of ${\textstyle K}$ is defined to be the function ${\textstyle p_{K}:X\to [0,\infty ],}$ valued in the extended real numbers, defined by $${\displaystyle p_{K}(x):=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\}\quad {\text{ for every }}x\in X,}$$ where the infimum of the empty set is defined to be positive infinity ${\textstyle \,\infty \,}$ (which is not a real number so that ${\textstyle p_{K}(x)}$ would then not be real-valued).

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

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

(where all three of these sets are necessarily absorbing in ${\textstyle X}$ and the first and last are also disks).

Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can be associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis. In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of ${\textstyle X}$ into certain algebraic properties of a function on ${\textstyle X.}$

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

Definition

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

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

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

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

Some conditions making a gauge real-valued

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

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

Motivating examples

Example 1

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

Example 2

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

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

Therefore, ${\textstyle p_{K}}$ is a seminorm on ${\textstyle X,}$ with an induced topology. This is characteristic of Minkowski functionals defined via "nice" sets. There is a one-to-one correspondence between seminorms and the Minkowski functional given by such sets. What is meant precisely by "nice" is discussed in the section below.

Notice that, in contrast to a stronger requirement for a norm, ${\textstyle p_{K}(x)=0}$ need not imply ${\textstyle x=0.}$ In the above example, one can take a nonzero ${\textstyle x}$ from the kernel of ${\textstyle f.}$ Consequently, the resulting topology need not be Hausdorff.

Common conditions guaranteeing gauges are seminorms

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

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

Theorem ^[2]  — If ${\textstyle K}$ is an absorbing disk in a vector space ${\textstyle X}$ then the Minkowski functional of ${\textstyle K,}$ which is the map ${\textstyle p_{K}:X\to [0,\infty )}$ defined by $${\displaystyle p_{K}(x):=\inf\{r>0:x\in rK\},}$$ is a seminorm on ${\textstyle X.}$ Moreover, $${\displaystyle p_{K}(x)={\frac {1}{\sup\{r>0:rx\in K\}}}.}$$

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

Gauges of absorbing disks

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

Theorem — Assume that ${\textstyle K}$ is an absorbing subset of ${\textstyle X.}$ It is shown that:

1. If ${\textstyle K}$ is convex then ${\textstyle p_{K}}$ is subadditive.
2. If ${\textstyle K}$ is balanced then ${\textstyle p_{K}}$ is absolutely homogeneous; that is, ${\textstyle p_{K}(sx)=|s|p_{K}(x)}$ for all scalars ${\textstyle s.}$

Proof that the Gauge of an absorbing disk is a seminorm

Convexity and subadditivity A simple geometric argument that shows convexity of ${\textstyle K}$ implies subadditivity is as follows. Suppose for the moment that ${\textstyle p_{K}(x)=p_{K}(y)=r.}$ Then for all ${\textstyle e>0,}$${\textstyle x,y\in K_{e}:=(r,e)K.}$ Since ${\textstyle K}$ is convex and ${\textstyle r+e\neq 0,}$${\textstyle K_{e}}$ is also convex. Therefore, ${\textstyle {\frac {1}{2}}x+{\frac {1}{2}}y\in K_{e}.}$ By definition of the Minkowski functional ${\textstyle p_{K},}$$${\displaystyle p_{K}\left({\frac {1}{2}}x+{\frac {1}{2}}y\right)\leq r+e={\frac {1}{2}}p_{K}(x)+{\frac {1}{2}}p_{K}(y)+e.}$$ But the left hand side is ${\textstyle {\frac {1}{2}}p_{K}(x+y),}$ so that $${\displaystyle p_{K}(x+y)\leq p_{K}(x)+p_{K}(y)+2e.}$$ Since ${\textstyle e>0}$ was arbitrary, it follows that ${\textstyle p_{K}(x+y)\leq p_{K}(x)+p_{K}(y),}$ which is the desired inequality. The general case ${\textstyle p_{K}(x)>p_{K}(y)}$ is obtained after the obvious modification. Convexity of ${\textstyle K,}$ together with the initial assumption that the set ${\textstyle \{r>0:x\in rK\}}$ is nonempty, implies that ${\textstyle K}$ is absorbing. Balancedness and absolute homogeneity Notice that ${\textstyle K}$ being balanced implies that $${\displaystyle \lambda x\in rK\quad {\mbox{if and only if}}\quad x\in {\frac {r}{|\lambda |}}K.}$$ Therefore $${\displaystyle p_{K}(\lambda x)=\inf \left\{r>0:\lambda x\in rK\right\}=\inf \left\{r>0:x\in {\frac {r}{|\lambda |}}K\right\}=\inf \left\{|\lambda |{\frac {r}{|\lambda |}}>0:x\in {\frac {r}{|\lambda |}}K\right\}=|\lambda |p_{K}(x).}$$

Algebraic properties

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

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

Topological properties

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

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

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

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

Minimal requirements on the set

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

Properties

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

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

Summary — Suppose that ${\textstyle K}$ is a subset of a real or complex vector space ${\textstyle X.}$

1. Strict positive homogeneity : ${\textstyle p_{K}(rx)=rp_{K}(x)}$ for all ${\textstyle x\in X}$ and all positive real ${\textstyle r>0.}$
   • Positive/Nonnegative homogeneity : ${\textstyle p_{K}}$ is nonnegative homogeneous if and only if ${\textstyle p_{K}}$ is real-valued.
     • A map ${\textstyle p}$ is called nonnegative homogeneous ^[7] if ${\textstyle p(rx)=rp(x)}$ for all ${\textstyle x\in X}$ and all nonnegative real ${\textstyle r\geq 0.}$ Since ${\textstyle 0\cdot \infty }$ is undefined, a map that takes infinity as a value is not nonnegative homogeneous.
2. Real-values: ${\textstyle (0,\infty )K}$ is the set of all points on which ${\textstyle p_{K}}$ is real valued. So ${\textstyle p_{K}}$ is real-valued if and only if ${\textstyle (0,\infty )K=X,}$ in which case ${\textstyle 0\in K.}$
   • Value at ${\textstyle 0}$: ${\textstyle p_{K}(0)\neq \infty }$ if and only if ${\textstyle 0\in K}$ if and only if ${\textstyle p_{K}(0)=0.}$
   • Null space : If ${\textstyle x\in X}$ then ${\textstyle p_{K}(x)=0}$ if and only if ${\textstyle (0,\infty )x\subseteq (0,1)K}$ if and only if there exists a divergent sequence of positive real numbers ${\textstyle t_{1},t_{2},t_{3},\cdots \to \infty }$ such that ${\textstyle t_{n}x\in K}$ for all ${\textstyle n.}$ Moreover, the zero set of ${\textstyle p_{K}}$ is ${\textstyle \ker p_{K}~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{y\in X:p_{K}(y)=0\right\}={\textstyle \bigcap \limits _{e>0}}(0,e)K.}$
3. Comparison to a constant: If ${\textstyle 0\leq r\leq \infty }$ then for any ${\textstyle x\in X,}$${\textstyle p_{K}(x)<r}$ if and only if ${\textstyle x\in (0,r)K;}$ this can be restated as: If ${\textstyle 0\leq r\leq \infty }$ then ${\textstyle p_{K}^{-1}([0,r))=(0,r)K.}$
   • It follows that if ${\textstyle 0\leq R<\infty }$ is real then ${\textstyle p_{K}^{-1}([0,R])={\textstyle \bigcap \limits _{e>0}}(0,R+e)K,}$ where the set on the right hand side denotes ${\textstyle {\textstyle \bigcap \limits _{e>0}}[(0,R+e)K]}$ and not its subset ${\textstyle \left[{\textstyle \bigcap \limits _{e>0}}(0,R+e)\right]K=(0,R]K.}$ If ${\textstyle R>0}$ then these sets are equal if and only if ${\textstyle K}$ contains ${\textstyle \left\{y\in X:p_{K}(y)=1\right\}.}$
   • In particular, if ${\textstyle x\in RK}$ or ${\textstyle x\in (0,R]K}$ then ${\textstyle p_{K}(x)\leq R,}$ but importantly, the converse is not necessarily true.
4. Gauge comparison: For any subset ${\textstyle L\subseteq X,}$${\textstyle p_{K}\leq p_{L}}$ if and only if ${\textstyle (0,1)L\subseteq (0,1)K;}$ thus ${\textstyle p_{L}=p_{K}}$ if and only if ${\textstyle (0,1)L=(0,1)K.}$
   • The assignment ${\textstyle L\mapsto p_{L}}$ is order-reversing in the sense that if ${\textstyle K\subseteq L}$ then ${\textstyle p_{L}\leq p_{K}.}$ ^[8]
   • Because the set ${\textstyle L:=(0,1)K}$ satisfies ${\textstyle (0,1)L=(0,1)K,}$ it follows that replacing ${\textstyle K}$ with ${\textstyle p_{K}^{-1}([0,1))=(0,1)K}$ will not change the resulting Minkowski functional. The same is true of ${\textstyle L:=(0,1]K}$ and of ${\textstyle L:=p_{K}^{-1}([0,1]).}$
   • If ${\textstyle D~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~\left\{y\in X:p_{K}(y)=1{\text{ or }}p_{K}(y)=0\right\}}$ then ${\textstyle p_{D}=p_{K}}$ and ${\textstyle D}$ has the particularly nice property that if ${\textstyle r>0}$ is real then ${\textstyle x\in rD}$ if and only if ${\textstyle p_{D}(x)=r}$ or ${\textstyle p_{D}(x)=0.}$ ^{[note 1]} Moreover, if ${\textstyle r>0}$ is real then ${\textstyle p_{D}(x)\leq r}$ if and only if ${\textstyle x\in (0,r]D.}$
5. Subadditive /Triangle inequality: ${\textstyle p_{K}}$ is subadditive if and only if ${\textstyle (0,1)K}$ is convex. If ${\textstyle K}$ is convex then so are both ${\textstyle (0,1)K}$ and ${\textstyle (0,1]K}$ and moreover, ${\textstyle p_{K}}$ is subadditive.
6. Scaling the set: If ${\textstyle s\neq 0}$ is a scalar then ${\textstyle p_{sK}(y)=p_{K}\left({\tfrac {1}{s}}y\right)}$ for all ${\textstyle y\in X.}$ Thus if ${\textstyle 0<r<\infty }$ is real then ${\textstyle p_{rK}(y)=p_{K}\left({\tfrac {1}{r}}y\right)={\tfrac {1}{r}}p_{K}(y).}$
7. Symmetric: ${\textstyle p_{K}}$ is symmetric (meaning that ${\textstyle p_{K}(-y)=p_{K}(y)}$ for all ${\textstyle y\in X}$) if and only if ${\textstyle (0,1)K}$ is a symmetric set (meaning that${\textstyle (0,1)K=-(0,1)K}$), which happens if and only if ${\textstyle p_{K}=p_{-K}.}$
8. Absolute homogeneity : ${\textstyle p_{K}(ux)=p_{K}(x)}$ for all ${\textstyle x\in X}$ and all unit length scalars ${\textstyle u}$ ^{[note 2]} if and only if ${\textstyle (0,1)uK\subseteq (0,1)K}$ for all unit length scalars ${\textstyle u,}$ in which case ${\textstyle p_{K}(sx)=|s|p_{K}(x)}$ for all ${\textstyle x\in X}$ and all non-zero scalars ${\textstyle s\neq 0.}$ If in addition ${\textstyle p_{K}}$ is also real-valued then this holds for all scalars ${\textstyle s}$ (that is, ${\textstyle p_{K}}$ is absolutely homogeneous ^{[note 3]} ).
   • ${\textstyle (0,1)uK\subseteq (0,1)K}$ for all unit length ${\textstyle u}$ if and only if ${\textstyle (0,1)uK=(0,1)K}$ for all unit length ${\textstyle u.}$
   • ${\textstyle sK\subseteq K}$ for all unit scalars ${\textstyle s}$ if and only if ${\textstyle sK=K}$ for all unit scalars ${\textstyle s;}$ if this is the case then ${\textstyle (0,1)K=(0,1)sK}$ for all unit scalars ${\textstyle s.}$
   • The Minkowski functional of any balanced set is a balanced function. ^[8]
9. Absorbing : If ${\textstyle K}$ is convex or balanced and if ${\textstyle (0,\infty )K=X}$ then ${\textstyle K}$ is absorbing in ${\textstyle X.}$
   • If a set ${\textstyle A}$ is absorbing in ${\textstyle X}$ and ${\textstyle A\subseteq K}$ then ${\textstyle K}$ is absorbing in ${\textstyle X.}$
   • If ${\textstyle K}$ is convex and ${\textstyle 0\in K}$ then ${\textstyle [0,1]K=K,}$ in which case ${\textstyle (0,1)K\subseteq K.}$
10. Restriction to a vector subspace: If ${\textstyle S}$ is a vector subspace of ${\textstyle X}$ and if ${\textstyle p_{K\cap S}:S\to [0,\infty ]}$ denotes the Minkowski functional of ${\textstyle K\cap S}$ on ${\textstyle S,}$ then ${\textstyle p_{K}{\big \vert }_{S}=p_{K\cap S},}$ where ${\textstyle p_{K}{\big \vert }_{S}}$ denotes the restriction of ${\textstyle p_{K}}$ to ${\textstyle S.}$

Proof

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 ${\textstyle A\subseteq X}$ that satisfies ${\textstyle (0,\infty )A=X}$ is necessarily absorbing in ${\textstyle X}$ is straightforward and can be found in the article on absorbing sets. For any real ${\textstyle t>0,}$ $${\displaystyle \{r>0:tx\in rK\}=\{t(r/t):x\in (r/t)K\}=t\{s>0:x\in sK\}}$$ so that taking the infimum of both sides shows that $${\displaystyle p_{K}(tx)=\inf\{r>0:tx\in rK\}=t\inf\{s>0:x\in sK\}=tp_{K}(x).}$$ This proves that Minkowski functionals are strictly positive homogeneous. For ${\textstyle 0\cdot p_{K}(x)}$ to be well-defined, it is necessary and sufficient that ${\textstyle p_{K}(x)\neq \infty$ ;} thus ${\textstyle p_{K}(tx)=tp_{K}(x)}$ for all ${\textstyle x\in X}$ and all non-negative real ${\textstyle t\geq 0}$ if and only if ${\textstyle p_{K}}$ is real-valued. The hypothesis of statement (7) allows us to conclude that ${\textstyle p_{K}(sx)=p_{K}(x)}$ for all ${\textstyle x\in X}$ and all scalars ${\textstyle s}$ satisfying ${\textstyle |s|=1.}$ Every scalar ${\textstyle s}$ is of the form ${\textstyle re^{it}}$ for some real ${\textstyle t}$ where ${\textstyle r:=|s|\geq 0}$ and ${\textstyle e^{it}}$ is real if and only if ${\textstyle s}$ is real. The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of ${\textstyle p_{K},}$ and from the positive homogeneity of ${\textstyle p_{K}}$ when ${\textstyle p_{K}}$ is real-valued. ${\textstyle \blacksquare }$

Examples

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

$${\displaystyle \left\{x\in X:p_{L}(x)<1{\text{ for all }}L\in {\mathcal {L}}\right\}\quad \subseteq \quad I\quad \subseteq \quad \left\{x\in X:p_{L}(x)\leq 1{\text{ for all }}L\in {\mathcal {L}}\right\}}$$ then ${\textstyle p_{I}(x)=\sup \left\{p_{L}(x):L\in {\mathcal {L}}\right\}}$ for all ${\textstyle x\in X.}$

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

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

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

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

Positive homogeneity characterizes Minkowski functionals

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

Theorem — Let ${\textstyle f:X\to [0,\infty ]}$ be any function. The following statements are equivalent:

1. Strict positive homogeneity : ${\textstyle \;f(tx)=tf(x)}$ for all ${\textstyle x\in X}$ and all positive real ${\textstyle t>0.}$
   • This statement is equivalent to: ${\textstyle f(tx)\leq tf(x)}$ for all ${\textstyle x\in X}$ and all positive real ${\textstyle t>0.}$
2. ${\textstyle f}$ is a Minkowski functional: meaning that there exists a subset ${\textstyle S\subseteq X}$ such that ${\textstyle f=p_{S}.}$
3. ${\textstyle f=p_{K}}$ where ${\textstyle K:=\{x\in X:f(x)\leq 1\}.}$
4. ${\textstyle f=p_{V}\,}$ where ${\textstyle V\,:=\{x\in X:f(x)<1\}.}$

Moreover, if ${\textstyle f}$ never takes on the value ${\textstyle \,\infty \,}$ (so that the product ${\textstyle 0\cdot f(x)}$ is always well-defined) then this list may be extended to include:

5. Positive / Nonnegative homogeneity : ${\textstyle f(tx)=tf(x)}$ for all ${\textstyle x\in X}$ and all nonnegative real ${\textstyle t\geq 0}$.

Proof

If ${\textstyle f(tx)\leq tf(x)}$ holds for all ${\textstyle x\in X}$ and real ${\textstyle t>0}$ then ${\textstyle tf(x)=tf\left({\tfrac {1}{t}}(tx)\right)\leq t{\tfrac {1}{t}}f(tx)=f(tx)\leq tf(x)}$ so that ${\textstyle tf(x)=f(tx).}$ 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 ${\textstyle f:X\to [0,\infty ]}$ is a function such that ${\textstyle f(tx)=tf(x)}$ for all ${\textstyle x\in X}$ and all real ${\textstyle t>0}$ and let ${\textstyle K:=\{y\in X:f(y)\leq 1\}.}$ For all real ${\textstyle t>0,}$${\textstyle f(0)=f(t0)=tf(0)}$ so by taking ${\textstyle t=2}$ for instance, it follows that either ${\textstyle f(0)=0}$ or ${\textstyle f(0)=\infty .}$ Let ${\textstyle x\in X.}$ It remains to show that ${\textstyle f(x)=p_{K}(x).}$ It will now be shown that if ${\textstyle f(x)=0}$ or ${\textstyle f(x)=\infty }$ then ${\textstyle f(x)=p_{K}(x),}$ so that in particular, it will follow that ${\textstyle f(0)=p_{K}(0).}$ So suppose that ${\textstyle f(x)=0}$ or ${\textstyle f(x)=\infty$ ;} in either case ${\textstyle f(tx)=tf(x)=f(x)}$ for all real ${\textstyle t>0.}$ Now if ${\textstyle f(x)=0}$ then this implies that that ${\textstyle tx\in K}$ for all real ${\textstyle t>0}$ (since ${\textstyle f(tx)=0\leq 1}$), which implies that ${\textstyle p_{K}(x)=0,}$ as desired. Similarly, if ${\textstyle f(x)=\infty }$ then ${\textstyle tx\not \in K}$ for all real ${\textstyle t>0,}$ which implies that ${\textstyle p_{K}(x)=\infty ,}$ as desired. Thus, it will henceforth be assumed that ${\textstyle R:=f(x)}$ a positive real number and that ${\textstyle x\neq 0}$ (importantly, however, the possibility that ${\textstyle p_{K}(x)}$ is ${\textstyle 0}$ or ${\textstyle \,\infty \,}$ has not yet been ruled out). Recall that just like ${\textstyle f,}$ the function ${\textstyle p_{K}}$ satisfies ${\textstyle p_{K}(tx)=tp_{K}(x)}$ for all real ${\textstyle t>0.}$ Since ${\textstyle 0<{\tfrac {1}{R}}<\infty ,}$${\textstyle p_{K}(x)=R=f(x)}$ if and only if ${\textstyle p_{K}\left({\tfrac {1}{R}}x\right)=1=f\left({\tfrac {1}{R}}x\right)}$ so assume without loss of generality that ${\textstyle R=1}$ and it remains to show that ${\textstyle p_{K}\left({\tfrac {1}{R}}x\right)=1.}$ Since ${\textstyle f(x)=1,}$${\textstyle x\in K\subseteq (0,1]K,}$ which implies that ${\textstyle p_{K}(x)\leq 1}$ (so in particular, ${\textstyle p_{K}(x)\neq \infty }$ is guaranteed). It remains to show that ${\textstyle p_{K}(x)\geq 1,}$ which recall happens if and only if ${\textstyle x\not \in (0,1)K.}$ So assume for the sake of contradiction that ${\textstyle x\in (0,1)K}$ and let ${\textstyle 0<r<1}$ and ${\textstyle k\in K}$ be such that ${\textstyle x=rk,}$ where note that ${\textstyle k\in K}$ implies that ${\textstyle f(k)\leq 1.}$ Then ${\textstyle 1=f(x)=f(rk)=rf(k)\leq r<1.}$${\textstyle \blacksquare }$

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 ${\textstyle f:X\to \mathbb {R} }$ (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 ^[9]  — Let ${\textstyle f:X\to [0,\infty ]}$ be any function and ${\textstyle K\subseteq X}$ be any subset. The following statements are equivalent:

1. ${\textstyle f}$ is (strictly) positive homogeneous, ${\textstyle f(0)=0,}$ and

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

2. ${\textstyle f}$ is the Minkowski functional of ${\textstyle K}$ (that is, ${\textstyle f=p_{K}}$), ${\textstyle K}$ contains the origin, and ${\textstyle K}$ is star-shaped at the origin.
   • The set ${\textstyle K}$ is star-shaped at the origin if and only if ${\textstyle tk\in K}$ whenever ${\textstyle k\in K}$ and ${\textstyle 0\leq t\leq 1.}$ A set that is star-shaped at the origin is sometimes called a star set. ^[10]

Characterizing Minkowski functionals that are seminorms

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

Theorem — Let ${\textstyle K}$ be a subset of a real or complex vector space ${\textstyle X.}$ Then ${\textstyle p_{K}}$ is a seminorm on ${\textstyle X}$ if and only if all of the following conditions hold:

1. ${\textstyle (0,\infty )K=X}$ (or equivalently, ${\textstyle p_{K}}$ is real-valued).
2. ${\textstyle (0,1)K}$ is convex (or equivalently, ${\textstyle p_{K}}$ is subadditive).
   • It suffices (but is not necessary) for ${\textstyle K}$ to be convex.
3. ${\textstyle (0,1)uK\subseteq (0,1)K}$ for all unit scalars ${\textstyle u.}$
   • This condition is satisfied if ${\textstyle K}$ is balanced or more generally if ${\textstyle uK\subseteq K}$ for all unit scalars ${\textstyle u.}$

in which case ${\textstyle 0\in K}$ and both ${\textstyle (0,1)K=\{x\in X:p(x)<1\}}$ and ${\textstyle \bigcap _{e>0}(0,1+e)K=\left\{x\in X:p_{K}(x)\leq 1\right\}}$ will be convex, balanced, and absorbing subsets of ${\textstyle X.}$

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

Corollary — If ${\textstyle K}$ is a convex, balanced, and absorbing subset of a real or complex vector space ${\textstyle X,}$ then ${\textstyle p_{K}}$ is a seminorm on ${\textstyle X.}$

Positive sublinear functions and Minkowski functionals

It may be shown that a real-valued subadditive function ${\textstyle f:X\to \mathbb {R} }$ on an arbitrary topological vector space ${\textstyle X}$ is continuous at the origin if and only if it is uniformly continuous, where if in addition ${\textstyle f}$ is nonnegative, then ${\textstyle f}$ is continuous if and only if ${\textstyle V:=\{x\in X:f(x)<1\}}$ is an open neighborhood in ${\textstyle X.}$ ^[11] If ${\textstyle f:X\to \mathbb {R} }$ is subadditive and satisfies ${\textstyle f(0)=0,}$ then ${\textstyle f}$ is continuous if and only if its absolute value ${\textstyle |f|:X\to [0,\infty )}$ is continuous.

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

Correspondence between open convex sets and positive continuous sublinear functions

Theorem ^[11]  — Suppose that ${\textstyle X}$ 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 ${\textstyle X}$ are exactly those sets that are of the form ${\textstyle z+\{x\in X:p(x)<1\}=\{x\in X:p(x-z)<1\}}$ for some ${\textstyle z\in X}$ and some positive continuous sublinear function ${\textstyle p}$ on ${\textstyle X.}$

Proof

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

See also

• Asymmetric norm  – Generalization of the concept of a norm
• Auxiliary normed space
• Cauchy's functional equation  – Functional equation
• Finest locally convex topology  – A vector space with a topology defined by convex open setsPages displaying short descriptions of redirect targets
• Finsler manifold  – Generalization of Riemannian manifolds
• Hadwiger's theorem  – Theorem in integral geometry
• Hugo Hadwiger  – Swiss mathematician (1908–1981)
• Locally convex topological vector space  – A vector space with a topology defined by convex open sets
• Morphological image processing  – Theory and technique for handling geometrical structuresPages displaying short descriptions of redirect targets
• Norm (mathematics)  – Length in a vector space
• Seminorm  – Mathematical function
• Topological vector space  – Vector space with a notion of nearness

Notes

1. It is in general false that ${\textstyle x\in rD}$ if and only if ${\textstyle p_{D}(x)=r}$ (for example, consider when ${\textstyle p_{K}}$ is a norm or a seminorm). The correct statement is: If ${\textstyle 0<r<\infty }$ then ${\textstyle x\in rD}$ if and only if ${\textstyle p_{D}(x)=r}$ or ${\textstyle p_{D}(x)=0.}$
2. ${\textstyle u}$ is having unit length means that ${\textstyle |u|=1.}$
3. The map ${\textstyle p_{K}}$ is called absolutely homogeneous if ${\textstyle |s|p_{K}(x)}$ is well-defined and ${\textstyle p_{K}(sx)=|s|p_{K}(x)}$ for all ${\textstyle x\in X}$ and all scalars ${\textstyle s}$ (not just non-zero scalars).

References

1. Narici & Beckenstein 2011, p. 109.
2. Narici & Beckenstein 2011, p. 119.
3. Jarchow 1981, pp. 104–108.
4. Narici & Beckenstein 2011, pp. 115–154.
5. Schaefer 1999, p. 40.
6. Narici & Beckenstein 2011, p. 119-120.
7. Kubrusly 2011, p. 200.
8. Schechter 1996, p. 316.
9. Schechter 1996, pp. 313–317.
10. Schechter 1996, p. 303.
11. Narici & Beckenstein 2011, pp. 192–193.

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Further reading

• F. Simeski, A. M. P. Boelens, and M. Ihme. "Modeling Adsorption in Silica Pores via Minkowski Functionals and Molecular Electrostatic Moments". Energies13 (22) 5976 (2020). doi : 10.3390/en13225976.