Minkowski functional

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

$$p\_K(x)\; :=\; \backslash inf\; \backslash \{r\; >\; 0\; :\; x\; \backslash in\; r\; K\backslash \},$$

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

For any $x\; \backslash in\; X,$ $p\_K(x)\; \backslash neq\; \backslash infty$ if and only if $\backslash \{r\; >\; 0\; :\; x\; \backslash in\; r\; K\backslash \}$ is not empty.

The arithmetic operations on $\backslash R$ can be extended to operate on $\backslash pm\; \backslash infty,$ where $\backslash frac\{r\}\{\backslash pm\; \backslash infty\}\; :=\; 0$ for all non-zero real

The products $0\; \backslash cdot\; \backslash infty$ and $0\; \backslash cdot\; -\; \backslash infty$ remain undefined.

In the field of convex analysis, the map $p\_K$ taking on the value of $\backslash ,\backslash infty\backslash ,$ is not necessarily an issue.

However, in functional analysis $p\_K$ is almost always real-valued (that is, to never take on the value of $\backslash ,\backslash infty\backslash ,$), which happens if and only if the set $\backslash \{r\; >\; 0\; :\; x\; \backslash in\; r\; K\backslash \}$ is non-empty for every $x\; \backslash in\; X.$

In order for $p\_K$ to be real-valued, it suffices for the origin of $X$ to belong to the {{em|algebraic interior}} or {{em|core}} of $K$ in $X.${{sfn|Narici|Beckenstein|2011|p=109}}

If $K$ is absorbing in $X,$ where recall that this implies that $0\; \backslash in\; K,$ then the origin belongs to the algebraic interior of $K$ in $X$ and thus $p\_K$ is real-valued.

Characterizations of when $p\_K$ is real-valued are given below.

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

Let $X$ be a vector space without topology with underlying scalar field $\backslash mathbb\{K\}.$

Let $f\; :\; X\; \backslash to\; \backslash mathbb\{K\}$ be any linear functional on $X$ (not necessarily continuous).

Fix $a\; >\; 0.$

Let $K$ be the set

$$K\; :=\; \backslash \{x\; \backslash in\; X\; :\; |f(x)|\; \backslash leq\; a\backslash \}$$

and let $p\_K$ be the Minkowski functional of $K.$

Then

$$p\_K(x)\; =\; \backslash frac\{1\}\{a\}\; |f(x)|\; \backslash quad\; \backslash text\{\; for\; all\; \}\; x\; \backslash in\; X.$$

The function $p\_K$ has the following properties:

1. It is {{em|subadditive}}: $p\_K(x\; +\; y)\; \backslash leq\; p\_K(x)\; +\; p\_K(y).$
2. It is {{em|absolutely homogeneous}}: $p\_K(s\; x)\; =\; |s|\; p\_K(x)$ for all scalars $s.$
3. It is {{em|nonnegative}}: $p\_K\; \backslash geq\; 0.$

Therefore, $p\_K$ is a seminorm on $X,$ 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, $p\_K(x)\; =\; 0$ need not imply $x\; =\; 0.$

In the above example, one can take a nonzero $x$ from the kernel of $f.$

Consequently, the resulting topology need not be Hausdorff.

To guarantee that $p\_K(0)\; =\; 0,$ it will henceforth be assumed that $0\; \backslash in\; K.$

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

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|p=119}}|math_statement=

If $K$ is an absorbing disk in a vector space $X$ then the Minkowski functional of $K,$ which is the map $p\_K\; :\; X\; \backslash to\; [0,\; \backslash infty)$ defined by

$$p\_K(x)\; :=\; \backslash inf\; \backslash \{r\; >\; 0\; :\; x\; \backslash in\; r\; K\backslash \},$$

is a seminorm on $X.$

Moreover,

$$p\_K(x)\; =\; \backslash frac\{1\}\{\backslash sup\; \backslash \{r\; >\; 0\; :\; r\; x\; \backslash in\; K\backslash \}\}.$$

}}

More generally, if $K$ is convex and the origin belongs to the algebraic interior of $K,$ then $p\_K$ is a nonnegative sublinear functional on $X,$ which implies in particular that it is subadditive and positive homogeneous.

If $K$ is absorbing in $X$ then $p\_\{[0,\; 1]\; K\}$ is positive homogeneous, meaning that $p\_\{[0,\; 1]\; K\}(s\; x)\; =\; s\; p\_\{[0,\; 1]\; K\}(x)$ for all real $s\; \backslash geq\; 0,$ where $[0,\; 1]\; K\; =\; \backslash \{t\; k\; :\; t\; \backslash in\; [0,\; 1],\; k\; \backslash in\; K\backslash \}.${{sfn|Jarchow|1981|pp=104-108}}

If $q$ is a nonnegative real-valued function on $X$ that is positive homogeneous, then the sets and $D\; :=\; \backslash \{x\; \backslash in\; X\; :\; q(x)\; \backslash leq\; 1\backslash \}$ satisfy $[0,\; 1]\; U\; =\; U$ and $[0,\; 1]\; D\; =\; D;$

if in addition $q$ is absolutely homogeneous then both $U$ and $D$ are balanced.{{sfn|Jarchow|1981|pp=104-108}}

Arguably the most common requirements placed on a set $K$ to guarantee that $p\_K$ is a seminorm are that $K$ be an absorbing disk in $X.$

Due to how common these assumptions are, the properties of a Minkowski functional $p\_K$ when $K$ is an absorbing disk will now be investigated.

Since all of the results mentioned above made few (if any) assumptions on $K,$ they can be applied in this special case.

{{Math theorem|name=Theorem|math_statement=

Assume that $K$ is an absorbing subset of $X.$

It is shown that:

1. If $K$ is convex then $p\_K$ is subadditive.
2. If $K$ is balanced then $p\_K$ is absolutely homogeneous; that is, $p\_K(s\; x)\; =\; |s|\; p\_K(x)$ for all scalars $s.$

}}

{{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 $K$ implies subadditivity is as follows.

Suppose for the moment that $p\_K(x)\; =\; p\_K(y)\; =\; r.$

Then for all $e\; >\; 0,$ $x,\; y\; \backslash in\; K\_e\; :=\; (r,\; e)\; K.$

Since $K$ is convex and $r\; +\; e\; \backslash neq\; 0,$ $K\_e$ is also convex.

Therefore, $\backslash frac\{1\}\{2\}\; x\; +\; \backslash frac\{1\}\{2\}\; y\; \backslash in\; K\_e.$

By definition of the Minkowski functional $p\_K,$

$$p\_K\backslash left(\backslash frac\{1\}\{2\}\; x\; +\; \backslash frac\{1\}\{2\}\; y\backslash right)\; \backslash leq\; r\; +\; e\; =\; \backslash frac\{1\}\{2\}\; p\_K(x)\; +\; \backslash frac\{1\}\{2\}\; p\_K(y)\; +\; e.$$

But the left hand side is $\backslash frac\{1\}\{2\}\; p\_K(x\; +\; y),$ so that

$$p\_K(x\; +\; y)\; \backslash leq\; p\_K(x)\; +\; p\_K(y)\; +\; 2\; e.$$

Since $e\; >\; 0$ was arbitrary, it follows that $p\_K(x\; +\; y)\; \backslash leq\; p\_K(x)\; +\; p\_K(y),$ which is the desired inequality.

The general case $p\_K(x)\; >\; p\_K(y)$ is obtained after the obvious modification.

Convexity of $K,$ together with the initial assumption that the set $\backslash \{r\; >\; 0\; :\; x\; \backslash in\; r\; K\backslash \}$ is nonempty, implies that $K$ is absorbing.

Balancedness and absolute homogeneity

Notice that $K$ being balanced implies that

$$\backslash lambda\; x\; \backslash in\; r\; K\; \backslash quad\; \backslash mbox\{if\; and\; only\; if\}\; \backslash quad\; x\; \backslash in\; \backslash frac\{r\}$$

Therefore

$$p\_K\; (\backslash lambda\; x)\; =\; \backslash inf\; \backslash left\backslash \{r\; >\; 0\; :\; \backslash lambda\; x\; \backslash in\; r\; K\; \backslash right\backslash \}$$

= \inf \left\{r > 0 : x \in \frac{r}

= \inf \left\

= |\lambda| p_K(x).

{{collapse bottom}}

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

• $p\_K$ is a seminorm on $X.$
• $p\_K$ is a norm on $X$ if and only if $K$ does not contain a non-trivial vector subspace.{{sfn|Narici|Beckenstein|2011|pp=115-154}}
• $p\_\{s\; K\}\; =\; \backslash frac\{1\}$

  s

  p_K for any scalar $s\; \backslash neq\; 0.${{sfn|Narici|Beckenstein|2011|pp=115-154}}
• If $J$ is an absorbing disk in $X$ and $J\; \backslash subseteq\; K$ then $p\_K\; \backslash leq\; p\_J.$
• If $K$ is a set satisfying then $K$ is absorbing in $X$ and $p\; =\; p\_K,$ where $p\_K$ is the Minkowski functional associated with $K;$ that is, it is the gauge of $K.${{sfn|Schaefer|1999|p=40}}
• In particular, if $K$ is as above and $q$ is any seminorm on $X,$ then $q\; =\; p$ if and only if {{sfn|Schaefer|1999|p=40}}
• If $x\; \backslash in\; X$ satisfies then $x\; \backslash in\; K.$

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

where $\backslash operatorname\{Int\}\_X\; K$ is the topological interior and $\backslash operatorname\{Cl\}\_X\; K$ is the topological closure of $K$ in $X.${{sfn|Narici|Beckenstein|2011|p=119-120}}

Importantly, it was {{em|not}} assumed that $p\_K$ was continuous nor was it assumed that $K$ had any topological properties.

Moreover, the Minkowski functional $p\_K$ is continuous if and only if $K$ is a neighborhood of the origin in $X.${{sfn|Narici|Beckenstein|2011|p=119-120}}

If $p\_K$ is continuous then{{sfn|Narici|Beckenstein|2011|p=119-120}}

This section will investigate the most general case of the gauge of {{em|any}} subset $K$ of $X.$

The more common special case where $K$ is assumed to be an absorbing disk in $X$ was discussed above.

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

Throughout, $K$ is any subset of $X.$

{{Math theorem|name=Summary|style=overflow:scroll|math_statement=

Suppose that $K$ is a subset of a real or complex vector space $X.$

1. Strict positive homogeneity: $p\_K(r\; x)\; =\; r\; p\_K(x)$ for all $x\; \backslash in\; X$ and all {{em|positive}} real $r\; >\; 0.$
2. * Positive/Nonnegative homogeneity: $p\_K$ is nonnegative homogeneous if and only if $p\_K$ is real-valued.
3. ** A map $p$ is called {{em|nonnegative homogeneous}}{{sfn|Kubrusly|2011|p=200}} if $p(r\; x)\; =\; r\; p(x)$ for all $x\; \backslash in\; X$ and all {{em|nonnegative}} real $r\; \backslash geq\; 0.$ Since $0\; \backslash cdot\; \backslash infty$ is undefined, a map that takes infinity as a value is not nonnegative homogeneous.
4. Real-values: $(0,\; \backslash infty)\; K$ is the set of all points on which $p\_K$ is real valued. So $p\_K$ is real-valued if and only if $(0,\; \backslash infty)\; K\; =\; X,$ in which case $0\; \backslash in\; K.$
5. * Value at $0$: $p\_K(0)\; \backslash neq\; \backslash infty$ if and only if $0\; \backslash in\; K$ if and only if $p\_K(0)\; =\; 0.$
6. * Null space: If $x\; \backslash in\; X$ then $p\_K(x)\; =\; 0$ if and only if $(0,\; \backslash infty)\; x\; \backslash subseteq\; (0,\; 1)\; K$ if and only if there exists a divergent sequence of positive real numbers $t\_1,\; t\_2,\; t\_3,\; \backslash cdots\; \backslash to\; \backslash infty$ such that $t\_n\; x\; \backslash in\; K$ for all $n.$ Moreover, the zero set of $p\_K$ is $\backslash ker\; p\_K\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash left\backslash \{y\; \backslash in\; X\; :\; p\_K(y)\; =\; 0\; \backslash right\backslash \}\; =\; \{\backslash textstyle\backslash bigcap\backslash limits\_\{e\; >\; 0\}\}\; (0,\; e)\; K.$
7. Comparison to a constant: If $0\; \backslash leq\; r\; \backslash leq\; \backslash infty$ then for any $x\; \backslash in\; X,$ if and only if $x\; \backslash in\; (0,\; r)\; K;$ this can be restated as: If $0\; \backslash leq\; r\; \backslash leq\; \backslash infty$ then $p\_K^\{-1\}([0,\; r))\; =\; (0,\; r)\; K.$
8. * It follows that if is real then $p\_K^\{-1\}([0,\; R])\; =\; \{\backslash textstyle\backslash bigcap\backslash limits\_\{e\; >\; 0\}\}\; (0,\; R\; +\; e)\; K,$ where the set on the right hand side denotes $\{\backslash textstyle\backslash bigcap\backslash limits\_\{e\; >\; 0\}\}\; [(0,\; R\; +\; e)\; K]$ and not its subset $\backslash left[\{\backslash textstyle\backslash bigcap\backslash limits\_\{e\; >\; 0\}\}\; (0,\; R\; +\; e)\backslash right]\; K\; =\; (0,\; R]\; K.$ If $R\; >\; 0$ then these sets are equal if and only if $K$ contains $\backslash left\backslash \{y\; \backslash in\; X\; :\; p\_K(y)\; =\; 1\; \backslash right\backslash \}.$
9. * In particular, if $x\; \backslash in\; R\; K$ or $x\; \backslash in\; (0,\; R]\; K$ then $p\_K(x)\; \backslash leq\; R,$ but importantly, the converse is not necessarily true.
10. Gauge comparison: For any subset $L\; \backslash subseteq\; X,$ $p\_K\; \backslash leq\; p\_L$ if and only if $(0,\; 1)\; L\; \backslash subseteq\; (0,\; 1)\; K;$ thus $p\_L\; =\; p\_K$ if and only if $(0,\; 1)\; L\; =\; (0,\; 1)\; K.$
11. * The assignment $L\; \backslash mapsto\; p\_L$ is order-reversing in the sense that if $K\; \backslash subseteq\; L$ then $p\_L\; \backslash leq\; p\_K.${{sfn|Schechter|1996|p=316}}
12. * Because the set $L\; :=\; (0,\; 1)\; K$ satisfies $(0,\; 1)\; L\; =\; (0,\; 1)\; K,$ it follows that replacing $K$ with $p\_K^\{-1\}([0,\; 1))\; =\; (0,\; 1)\; K$ will not change the resulting Minkowski functional. The same is true of $L\; :=\; (0,\; 1]\; K$ and of $L\; :=\; p\_K^\{-1\}([0,\; 1]).$
13. * If $D\; ~\backslash stackrel\{\backslash scriptscriptstyle\backslash text\{def\}\}\{=\}~\; \backslash left\backslash \{y\; \backslash in\; X\; :\; p\_K(y)\; =\; 1\; \backslash text\{\; or\; \}\; p\_K(y)\; =\; 0\; \backslash right\backslash \}$ then $p\_D\; =\; p\_K$ and $D$ has the particularly nice property that if $r\; >\; 0$ is real then $x\; \backslash in\; r\; D$ if and only if $p\_D(x)\; =\; r$ or $p\_D(x)\; =\; 0.$It is in general {{em|false}} that $x\; \backslash in\; r\; D$ if and only if $p\_D(x)\; =\; r$ (for example, consider when $p\_K$ is a norm or a seminorm). The correct statement is: If then $x\; \backslash in\; r\; D$ if and only if $p\_D(x)\; =\; r$ or $p\_D(x)\; =\; 0.$ Moreover, if $r\; >\; 0$ is real then $p\_D(x)\; \backslash leq\; r$ if and only if $x\; \backslash in\; (0,\; r]\; D.$
14. Subadditive/Triangle inequality: $p\_K$ is subadditive if and only if $(0,\; 1)\; K$ is convex. If $K$ is convex then so are both $(0,\; 1)\; K$ and $(0,\; 1]\; K$ and moreover, $p\_K$ is subadditive.
15. Scaling the set: If $s\; \backslash neq\; 0$ is a scalar then $p\_\{s\; K\}(y)\; =\; p\_K\backslash left(\backslash tfrac\{1\}\{s\}\; y\backslash right)$ for all $y\; \backslash in\; X.$ Thus if is real then $p\_\{r\; K\}(y)\; =\; p\_K\backslash left(\backslash tfrac\{1\}\{r\}\; y\backslash right)\; =\; \backslash tfrac\{1\}\{r\}\; p\_K(y).$
16. Symmetric: $p\_K$ is symmetric (meaning that $p\_K(-\; y)\; =\; p\_K(y)$ for all $y\; \backslash in\; X$) if and only if $(0,\; 1)\; K$ is a symmetric set (meaning that$(0,\; 1)\; K\; =\; -\; (0,\; 1)\; K$), which happens if and only if $p\_K\; =\; p\_\{-\; K\}.$
17. Absolute homogeneity: $p\_K(u\; x)\; =\; p\_K(x)$ for all $x\; \backslash in\; X$ and all unit length scalars $u$$u$ is having unit length means that $|u|\; =\; 1.$ if and only if $(0,\; 1)\; u\; K\; \backslash subseteq\; (0,\; 1)\; K$ for all unit length scalars $u,$ in which case $p\_K(s\; x)\; =\; |s|\; p\_K(x)$ for all $x\; \backslash in\; X$ and all {{em|non-zero}} scalars $s\; \backslash neq\; 0.$ If in addition $p\_K$ is also real-valued then this holds for {{em|all}} scalars $s$ (that is, $p\_K$ is absolutely homogeneousThe map $p\_K$ is called {{em|absolutely homogeneous}} if $|s|\; p\_K(x)$ is well-defined and $p\_K(s\; x)\; =\; |s|\; p\_K(x)$ for all $x\; \backslash in\; X$ and all scalars $s$ (not just non-zero scalars).).
18. * $(0,\; 1)\; u\; K\; \backslash subseteq\; (0,\; 1)\; K$ for all unit length $u$ if and only if $(0,\; 1)\; u\; K\; =\; (0,\; 1)\; K$ for all unit length $u.$
19. * $s\; K\; \backslash subseteq\; K$ for all unit scalars $s$ if and only if $s\; K\; =\; K$ for all unit scalars $s;$ if this is the case then $(0,\; 1)\; K\; =\; (0,\; 1)\; s\; K$ for all unit scalars $s.$
20. * The Minkowski functional of any balanced set is a balanced function.{{sfn|Schechter|1996|p=316}}
21. Absorbing: If $K$ is convex {{em|or}} balanced and if $(0,\; \backslash infty)\; K\; =\; X$ then $K$ is absorbing in $X.$
22. * If a set $A$ is absorbing in $X$ and $A\; \backslash subseteq\; K$ then $K$ is absorbing in $X.$
23. * If $K$ is convex and $0\; \backslash in\; K$ then $[0,\; 1]\; K\; =\; K,$ in which case $(0,\; 1)\; K\; \backslash subseteq\; K.$
24. Restriction to a vector subspace: If $S$ is a vector subspace of $X$ and if $p\_\{K\; \backslash cap\; S\}\; :\; S\; \backslash to\; [0,\; \backslash infty]$ denotes the Minkowski functional of $K\; \backslash cap\; S$ on $S,$ then $p\_K\backslash big\backslash vert\_S\; =\; p\_\{K\; \backslash cap\; S\},$ where $p\_K\backslash big\backslash vert\_S$ denotes the restriction of $p\_K$ to $S.$

}}

{{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 $A\; \backslash subseteq\; X$ that satisfies $(0,\; \backslash infty)\; A\; =\; X$ is necessarily absorbing in $X$ is straightforward and can be found in the article on absorbing sets.

For any real $t\; >\; 0,$

$$\backslash \{r\; >\; 0\; :\; t\; x\; \backslash in\; r\; K\backslash \}\; =\; \backslash \{t(r/t)\; :\; x\; \backslash in\; (r/t)\; K\backslash \}\; =\; t\; \backslash \{s\; >\; 0\; :\; x\; \backslash in\; s\; K\backslash \}$$

so that taking the infimum of both sides shows that

$$p\_K(tx)\; =\; \backslash inf\; \backslash \{r\; >\; 0\; :\; t\; x\; \backslash in\; r\; K\backslash \}\; =\; t\; \backslash inf\; \backslash \{s\; >\; 0\; :\; x\; \backslash in\; s\; K\backslash \}\; =\; t\; p\_K(x).$$

This proves that Minkowski functionals are strictly positive homogeneous. For $0\; \backslash cdot\; p\_K(x)$ to be well-defined, it is necessary and sufficient that $p\_K(x)\; \backslash neq\; \backslash infty;$ thus $p\_K(tx)\; =\; t\; p\_K(x)$ for all $x\; \backslash in\; X$ and all {{em|non-negative}} real $t\; \backslash geq\; 0$ if and only if $p\_K$ is real-valued.

The hypothesis of statement (7) allows us to conclude that $p\_K(s\; x)\; =\; p\_K(x)$ for all $x\; \backslash in\; X$ and all scalars $s$ satisfying $|s|\; =\; 1.$

Every scalar $s$ is of the form $r\; e^\{i\; t\}$ for some real $t$ where $r\; :=\; |s|\; \backslash geq\; 0$ and $e^\{i\; t\}$ is real if and only if $s$ is real.

The results in the statement about absolute homogeneity follow immediately from the aforementioned conclusion, from the strict positive homogeneity of $p\_K,$ and from the positive homogeneity of $p\_K$ when $p\_K$ is real-valued.

$\backslash blacksquare$

{{collapse bottom}}

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

then $p\_I(x)\; =\; \backslash sup\; \backslash left\backslash \{p\_L(x)\; :\; L\; \backslash in\; \backslash mathcal\{L\}\backslash right\backslash \}$ for all $x\; \backslash in\; X.$

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

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

The next example shows that the containment can be proper when $R\; =\; 1;$ the example may be generalized to any real $R\; >\; 0.$

Assuming that $[0,\; 1]\; K\; \backslash subseteq\; K,$ the following example is representative of how it happens that $x\; \backslash in\; X$ satisfies $p\_K(x)\; =\; 1$ but $x\; \backslash not\backslash in\; (0,\; 1]\; K.$

Example: Let $x\; \backslash in\; X$ be non-zero and let $K\; =\; [0,\; 1)\; x$ so that $[0,\; 1]\; K\; =\; K$ and $x\; \backslash not\backslash in\; K.$

From $x\; \backslash not\backslash in\; (0,\; 1)\; K\; =\; K$ it follows that $p\_K(x)\; \backslash geq\; 1.$

That $p\_K(x)\; \backslash leq\; 1$ follows from observing that for every $e\; >\; 0,$ $(0,\; 1\; +\; e)\; K\; =\; [0,\; 1\; +\; e)([0,\; 1)\; x)\; =\; [0,\; 1\; +\; e)\; x,$ which contains $x.$

Thus $p\_K(x)\; =\; 1$ and $x\; \backslash in\; \{\backslash textstyle\backslash bigcap\backslash limits\_\{e\; >\; 0\}\}\; (0,\; 1\; +\; e)\; K.$

However, $(0,\; 1]\; K\; =\; (0,\; 1]([0,\; 1)\; x)\; =\; [0,\; 1)\; x\; =\; K$ so that $x\; \backslash not\backslash in\; (0,\; 1]\; K,$ as desired.

$\backslash blacksquare$

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

{{Math theorem|name=Theorem|math_statement=

Let $f\; :\; X\; \backslash to\; [0,\; \backslash infty]$ be any function.

The following statements are equivalent:

1. Strict positive homogeneity: $\backslash ;\; f(t\; x)\; =\; t\; f(x)$ for all $x\; \backslash in\; X$ and all {{em|positive}} real $t\; >\; 0.$
2. * This statement is equivalent to: $f(t\; x)\; \backslash leq\; t\; f(x)$ for all $x\; \backslash in\; X$ and all positive real $t\; >\; 0.$
3. $f$ is a Minkowski functional: meaning that there exists a subset $S\; \backslash subseteq\; X$ such that $f\; =\; p\_S.$
4. $f\; =\; p\_K$ where $K\; :=\; \backslash \{x\; \backslash in\; X\; :\; f(x)\; \backslash leq\; 1\backslash \}.$
5. $f\; =\; p\_V\; \backslash ,$ where

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

{{ordered list|start=5

|Positive/Nonnegative homogeneity: $f(t\; x)\; =\; t\; f(x)$ for all $x\; \backslash in\; X$ and all {{em|nonnegative}} real $t\; \backslash geq\; 0$.

}}

}}

{{collapse top|title=Proof|left=true}}

If $f(t\; x)\; \backslash leq\; t\; f(x)$ holds for all $x\; \backslash in\; X$ and real $t\; >\; 0$ then $t\; f(x)\; =\; t\; f\backslash left(\backslash tfrac\{1\}\{t\}(t\; x)\backslash right)\; \backslash leq\; t\; \backslash tfrac\{1\}\{t\}\; f(t\; x)\; =\; f(t\; x)\; \backslash leq\; t\; f(x)$ so that $t\; f(x)\; =\; f(t\; x).$

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 $f\; :\; X\; \backslash to\; [0,\; \backslash infty]$ is a function such that $f(t\; x)\; =\; t\; f(x)$ for all $x\; \backslash in\; X$ and all real $t\; >\; 0$ and let $K\; :=\; \backslash \{y\; \backslash in\; X\; :\; f(y)\; \backslash leq\; 1\backslash \}.$

For all real $t\; >\; 0,$ $f(0)\; =\; f(t\; 0)\; =\; t\; f(0)$ so by taking $t\; =\; 2$ for instance, it follows that either $f(0)\; =\; 0$ or $f(0)\; =\; \backslash infty.$

Let $x\; \backslash in\; X.$

It remains to show that $f(x)\; =\; p\_K(x).$

It will now be shown that if $f(x)\; =\; 0$ or $f(x)\; =\; \backslash infty$ then $f(x)\; =\; p\_K(x),$ so that in particular, it will follow that $f(0)\; =\; p\_K(0).$

So suppose that $f(x)\; =\; 0$ or $f(x)\; =\; \backslash infty;$ in either case $f(t\; x)\; =\; t\; f(x)\; =\; f(x)$ for all real $t\; >\; 0.$

Now if $f(x)\; =\; 0$ then this implies that that $t\; x\; \backslash in\; K$ for all real $t\; >\; 0$ (since $f(t\; x)\; =\; 0\; \backslash leq\; 1$), which implies that $p\_K(x)\; =\; 0,$ as desired.

Similarly, if $f(x)\; =\; \backslash infty$ then $t\; x\; \backslash not\backslash in\; K$ for all real $t\; >\; 0,$ which implies that $p\_K(x)\; =\; \backslash infty,$ as desired.

Thus, it will henceforth be assumed that $R\; :=\; f(x)$ a positive real number and that $x\; \backslash neq\; 0$ (importantly, however, the possibility that $p\_K(x)$ is $0$ or $\backslash ,\backslash infty\backslash ,$ has not yet been ruled out).

Recall that just like $f,$ the function $p\_K$ satisfies $p\_K(t\; x)\; =\; t\; p\_K(x)$ for all real $t\; >\; 0.$

Since $p\_K(x)=\; R\; =\; f(x)$ if and only if $p\_K\backslash left(\backslash tfrac\{1\}\{R\}\; x\backslash right)\; =\; 1\; =\; f\backslash left(\backslash tfrac\{1\}\{R\}\; x\backslash right)$ so assume without loss of generality that $R\; =\; 1$ and it remains to show that $p\_K\backslash left(\backslash tfrac\{1\}\{R\}\; x\backslash right)\; =\; 1.$

Since $f(x)\; =\; 1,$ $x\; \backslash in\; K\; \backslash subseteq\; (0,\; 1]\; K,$ which implies that $p\_K(x)\; \backslash leq\; 1$ (so in particular, $p\_K(x)\; \backslash neq\; \backslash infty$ is guaranteed).

It remains to show that $p\_K(x)\; \backslash geq\; 1,$ which recall happens if and only if $x\; \backslash not\backslash in\; (0,\; 1)\; K.$

So assume for the sake of contradiction that $x\; \backslash in\; (0,\; 1)\; K$ and let and $k\; \backslash in\; K$ be such that $x\; =\; r\; k,$ where note that $k\; \backslash in\; K$ implies that $f(k)\; \backslash leq\; 1.$

Then $\backslash blacksquare$

{{collapse bottom}}

This theorem can be extended to characterize certain classes of $[-\; \backslash infty,\; \backslash infty]$-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 $f\; :\; X\; \backslash to\; \backslash R$ (such as linear functionals) can be written in terms of a unique Minkowski functional having a certain property.

{{Math theorem|name=Proposition{{sfn|Schechter|1996|pp=313-317}}|style=overflow:scroll|math_statement=

Let $f\; :\; X\; \backslash to\; [0,\; \backslash infty]$ be any function and $K\; \backslash subseteq\; X$ be any subset.

The following statements are equivalent:

1. $f$ is (strictly) positive homogeneous, $f(0)\; =\; 0,$ and

2. $f$ is the Minkowski functional of $K$ (that is, $f\; =\; p\_K$), $K$ contains the origin, and $K$ is star-shaped at the origin.
3. * The set $K$ is star-shaped at the origin if and only if $t\; k\; \backslash in\; K$ whenever $k\; \backslash in\; K$ and $0\; \backslash leq\; t\; \backslash leq\; 1.$ A set that is star-shaped at the origin is sometimes called a {{em|star set}}.{{sfn|Schechter|1996|p=303}}

}}

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

The common requirement that $K$ be convex is also weakened to only requiring that $(0,\; 1)\; K$ be convex.

{{Math theorem|name=Theorem|math_statement=

Let $K$ be a subset of a real or complex vector space $X.$

Then $p\_K$ is a seminorm on $X$ if and only if all of the following conditions hold:

1. $(0,\; \backslash infty)\; K\; =\; X$ (or equivalently, $p\_K$ is real-valued).
2. $(0,\; 1)\; K$ is convex (or equivalently, $p\_K$ is subadditive).
3. * It suffices (but is not necessary) for $K$ to be convex.
4. $(0,\; 1)\; u\; K\; \backslash subseteq\; (0,\; 1)\; K$ for all unit scalars $u.$
5. * This condition is satisfied if $K$ is balanced or more generally if $u\; K\; \backslash subseteq\; K$ for all unit scalars $u.$

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

Conversely, if $f$ is a seminorm on $X$ then the set satisfies all three of the above conditions (and thus also the conclusions) and also $f\; =\; p\_V;$

moreover, $V$ is necessarily convex, balanced, absorbing, and satisfies $(0,\; 1)\; V\; =\; V\; =\; [0,\; 1]\; V.$

}}

{{Math theorem|name=Corollary|math_statement=

If $K$ is a convex, balanced, and absorbing subset of a real or complex vector space $X,$ then $p\_K$ is a seminorm on $X.$

}}

It may be shown that a real-valued subadditive function $f\; :\; X\; \backslash to\; \backslash R$ on an arbitrary topological vector space $X$ is continuous at the origin if and only if it is uniformly continuous, where if in addition $f$ is nonnegative, then $f$ is continuous if and only if is an open neighborhood in $X.${{sfn|Narici|Beckenstein|2011|pp=192-193}}

If $f\; :\; X\; \backslash to\; \backslash R$ is subadditive and satisfies $f(0)\; =\; 0,$ then $f$ is continuous if and only if its absolute value $|f|\; :\; X\; \backslash to\; [0,\; \backslash infty)$ is continuous.

A {{em|nonnegative sublinear function}} is a nonnegative homogeneous function $f\; :\; X\; \backslash to\; [0,\; \backslash infty)$ that satisfies the triangle inequality.

It follows immediately from the results below that for such a function $f,$ if then $f\; =\; p\_V.$

Given $K\; \backslash subseteq\; X,$ the Minkowski functional $p\_K$ is a sublinear function if and only if it is real-valued and subadditive, which is happens if and only if $(0,\; \backslash infty)\; K\; =\; X$ and $(0,\; 1)\; K$ is convex.

{{Math theorem|name=Theorem{{sfn|Narici|Beckenstein|2011|pp=192–193}}|style=overflow:scroll|math_statement=

Suppose that $X$ 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 $X$ are exactly those sets that are of the form for some $z\; \backslash in\; X$ and some positive continuous sublinear function $p$ on $X.$

}}

{{collapse top|title=Proof|left=true}}

Let $V\; \backslash neq\; \backslash varnothing$ be an open convex subset of $X.$

If $0\; \backslash in\; V$ then let $z\; :=\; 0$ and otherwise let $z\; \backslash in\; V$ be arbitrary.

Let $p\; =\; p\_K\; :\; X\; \backslash to\; [0,\; \backslash infty)$ be the Minkowski functional of $K\; :=\; V\; -\; z$ where this convex open neighborhood of the origin satisfies $(0,\; 1)\; K\; =\; K.$

Then $p$ is a continuous sublinear function on $X$ since $V\; -\; z$ is convex, absorbing, and open (however, $p$ is not necessarily a seminorm since it is not necessarily absolutely homogeneous).

From the properties of Minkowski functionals, we have $p\_K^\{-1\}([0,\; 1))\; =\; (0,\; 1)\; K,$ from which it follows that and so

Since this completes the proof. $\backslash blacksquare$

{{collapse bottom}}

• {{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}}

{{reflist|group=note}}

{{reflist|group=proof}}

{{reflist}}

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• {{Dineen Complex Analysis in Locally Convex Spaces}}
• {{Dunford Schwartz Linear Operators Part 1 General Theory}}
• {{Edwards Functional Analysis Theory and Applications}}
• {{Grothendieck Topological Vector Spaces}}
• {{Hogbe-Nlend Bornologies and Functional Analysis}}
• {{Hogbe-Nlend Moscatelli Nuclear and Conuclear Spaces}}
• {{Husain Khaleelulla Barrelledness in Topological and Ordered Vector Spaces}}
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• {{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}}
• {{Schechter Handbook of Analysis and Its Foundations}}
• {{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}}
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• 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). {{doi|10.3390/en13225976}}.

{{Functional analysis}}

{{Topological vector spaces}}

{{Convex analysis and variational analysis}}

Category:Convex analysis

Category:Functional analysis

Category:Hermann Minkowski