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Minkowski functional

From HandWiki - Reading time: 27 min
Short description: Function made from a set
Minkowski functional.svg

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

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

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

The Minkowski function is always non-negative (meaning [math]\displaystyle{ p_K \geq 0 }[/math]). 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, [math]\displaystyle{ p_K }[/math] might not be real-valued since for any given [math]\displaystyle{ x \in X, }[/math] the value [math]\displaystyle{ p_K(x) }[/math] is a real number if and only if [math]\displaystyle{ \{r \gt 0 : x \in r K\} }[/math] is not empty. Consequently, [math]\displaystyle{ K }[/math] is usually assumed to have properties (such as being absorbing in [math]\displaystyle{ X, }[/math] for instance) that will guarantee that [math]\displaystyle{ p_K }[/math] is real-valued.

Definition

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

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

Some conditions making a gauge real-valued

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

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

Motivating examples

Example 1

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

Example 2

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

  1. It is subadditive: [math]\displaystyle{ p_K(x + y) \leq p_K(x) + p_K(y). }[/math]
  2. It is absolutely homogeneous: [math]\displaystyle{ p_K(s x) = |s| p_K(x) }[/math] for all scalars [math]\displaystyle{ s. }[/math]
  3. It is nonnegative: [math]\displaystyle{ p_K \geq 0. }[/math]

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

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

Common conditions guaranteeing gauges are seminorms

To guarantee that [math]\displaystyle{ p_K(0) = 0, }[/math] it will henceforth be assumed that [math]\displaystyle{ 0 \in K. }[/math]

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

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

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

Gauges of absorbing disks

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

Theorem — Assume that [math]\displaystyle{ K }[/math] is an absorbing subset of [math]\displaystyle{ X. }[/math] It is shown that:

  1. If [math]\displaystyle{ K }[/math] is convex then [math]\displaystyle{ p_K }[/math] is subadditive.
  2. If [math]\displaystyle{ K }[/math] is balanced then [math]\displaystyle{ p_K }[/math] is absolutely homogeneous; that is, [math]\displaystyle{ p_K(s x) = |s| p_K(x) }[/math] for all scalars [math]\displaystyle{ s. }[/math]
Proof that the Gauge of an absorbing disk is a seminorm

Convexity and subadditivity

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

But the left hand side is [math]\displaystyle{ \frac{1}{2} p_K(x + y), }[/math] so that [math]\displaystyle{ p_K(x + y) \leq p_K(x) + p_K(y) + 2 e. }[/math]

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

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

Balancedness and absolute homogeneity

Notice that [math]\displaystyle{ K }[/math] being balanced implies that [math]\displaystyle{ \lambda x \in r K \quad \mbox{if and only if} \quad x \in \frac{r}{|\lambda|} K. }[/math]

Therefore [math]\displaystyle{ p_K (\lambda x) = \inf \left\{r \gt 0 : \lambda x \in r K \right\} = \inf \left\{r \gt 0 : x \in \frac{r}{|\lambda|} K \right\} = \inf \left\{|\lambda|\frac{r}{|\lambda|} \gt 0 : x \in \frac{r}{|\lambda|} K \right\} = |\lambda| p_K(x). }[/math]

Algebraic properties

Let [math]\displaystyle{ X }[/math] be a real or complex vector space and let [math]\displaystyle{ K }[/math] be an absorbing disk in [math]\displaystyle{ X. }[/math]

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

Topological properties

Assume that [math]\displaystyle{ X }[/math] is a (real or complex) topological vector space (TVS) (not necessarily Hausdorff or locally convex) and let [math]\displaystyle{ K }[/math] be an absorbing disk in [math]\displaystyle{ X. }[/math] Then [math]\displaystyle{ \operatorname{Int}_X K \; \subseteq \; \{x \in X : p_K(x) \lt 1\} \; \subseteq \; K \; \subseteq \; \{x \in X : p_K(x) \leq 1\} \; \subseteq \; \operatorname{Cl}_X K, }[/math] where [math]\displaystyle{ \operatorname{Int}_X K }[/math] is the topological interior and [math]\displaystyle{ \operatorname{Cl}_X K }[/math] is the topological closure of [math]\displaystyle{ K }[/math] in [math]\displaystyle{ X. }[/math][6] Importantly, it was not assumed that [math]\displaystyle{ p_K }[/math] was continuous nor was it assumed that [math]\displaystyle{ K }[/math] had any topological properties.

Moreover, the Minkowski functional [math]\displaystyle{ p_K }[/math] is continuous if and only if [math]\displaystyle{ K }[/math] is a neighborhood of the origin in [math]\displaystyle{ X. }[/math][6] If [math]\displaystyle{ p_K }[/math] is continuous then[6] [math]\displaystyle{ \operatorname{Int}_X K = \{x \in X : p_K(x) \lt 1\} \quad \text{ and } \quad \operatorname{Cl}_X K = \{x \in X : p_K(x) \leq 1\}. }[/math]

Minimal requirements on the set

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

Properties

All results in this section may be applied to the case where [math]\displaystyle{ K }[/math] is an absorbing disk.

Throughout, [math]\displaystyle{ K }[/math] is any subset of [math]\displaystyle{ X. }[/math]

Summary — Suppose that [math]\displaystyle{ K }[/math] is a subset of a real or complex vector space [math]\displaystyle{ X. }[/math]

  1. Strict positive homogeneity: [math]\displaystyle{ p_K(r x) = r p_K(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all positive real [math]\displaystyle{ r \gt 0. }[/math]
    • Positive/Nonnegative homogeneity: [math]\displaystyle{ p_K }[/math] is nonnegative homogeneous if and only if [math]\displaystyle{ p_K }[/math] is real-valued.
      • A map [math]\displaystyle{ p }[/math] is called nonnegative homogeneous[7] if [math]\displaystyle{ p(r x) = r p(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all nonnegative real [math]\displaystyle{ r \geq 0. }[/math] Since [math]\displaystyle{ 0 \cdot \infty }[/math] is undefined, a map that takes infinity as a value is not nonnegative homogeneous.
  2. Real-values: [math]\displaystyle{ (0, \infty) K }[/math] is the set of all points on which [math]\displaystyle{ p_K }[/math] is real valued. So [math]\displaystyle{ p_K }[/math] is real-valued if and only if [math]\displaystyle{ (0, \infty) K = X, }[/math] in which case [math]\displaystyle{ 0 \in K. }[/math]
    • Value at [math]\displaystyle{ 0 }[/math]: [math]\displaystyle{ p_K(0) \neq \infty }[/math] if and only if [math]\displaystyle{ 0 \in K }[/math] if and only if [math]\displaystyle{ p_K(0) = 0. }[/math]
    • Null space: If [math]\displaystyle{ x \in X }[/math] then [math]\displaystyle{ p_K(x) = 0 }[/math] if and only if [math]\displaystyle{ (0, \infty) x \subseteq (0, 1) K }[/math] if and only if there exists a divergent sequence of positive real numbers [math]\displaystyle{ t_1, t_2, t_3, \cdots \to \infty }[/math] such that [math]\displaystyle{ t_n x \in K }[/math] for all [math]\displaystyle{ n. }[/math] Moreover, the zero set of [math]\displaystyle{ p_K }[/math] is [math]\displaystyle{ \ker p_K ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{y \in X : p_K(y) = 0 \right\} = {\textstyle\bigcap\limits_{e \gt 0}} (0, e) K. }[/math]
  3. Comparison to a constant: If [math]\displaystyle{ 0 \leq r \leq \infty }[/math] then for any [math]\displaystyle{ x \in X, }[/math] [math]\displaystyle{ p_K(x) \lt r }[/math] if and only if [math]\displaystyle{ x \in (0, r) K; }[/math] this can be restated as: If [math]\displaystyle{ 0 \leq r \leq \infty }[/math] then [math]\displaystyle{ p_K^{-1}([0, r)) = (0, r) K. }[/math]
    • It follows that if [math]\displaystyle{ 0 \leq R \lt \infty }[/math] is real then [math]\displaystyle{ p_K^{-1}([0, R]) = {\textstyle\bigcap\limits_{e \gt 0}} (0, R + e) K, }[/math] where the set on the right hand side denotes [math]\displaystyle{ {\textstyle\bigcap\limits_{e \gt 0}} [(0, R + e) K] }[/math] and not its subset [math]\displaystyle{ \left[{\textstyle\bigcap\limits_{e \gt 0}} (0, R + e)\right] K = (0, R] K. }[/math] If [math]\displaystyle{ R \gt 0 }[/math] then these sets are equal if and only if [math]\displaystyle{ K }[/math] contains [math]\displaystyle{ \left\{y \in X : p_K(y) = 1 \right\}. }[/math]
    • In particular, if [math]\displaystyle{ x \in R K }[/math] or [math]\displaystyle{ x \in (0, R] K }[/math] then [math]\displaystyle{ p_K(x) \leq R, }[/math] but importantly, the converse is not necessarily true.
  4. Gauge comparison: For any subset [math]\displaystyle{ L \subseteq X, }[/math] [math]\displaystyle{ p_K \leq p_L }[/math] if and only if [math]\displaystyle{ (0, 1) L \subseteq (0, 1) K; }[/math] thus [math]\displaystyle{ p_L = p_K }[/math] if and only if [math]\displaystyle{ (0, 1) L = (0, 1) K. }[/math]
    • The assignment [math]\displaystyle{ L \mapsto p_L }[/math] is order-reversing in the sense that if [math]\displaystyle{ K \subseteq L }[/math] then [math]\displaystyle{ p_L \leq p_K. }[/math][8]
    • Because the set [math]\displaystyle{ L := (0, 1) K }[/math] satisfies [math]\displaystyle{ (0, 1) L = (0, 1) K, }[/math] it follows that replacing [math]\displaystyle{ K }[/math] with [math]\displaystyle{ p_K^{-1}([0, 1)) = (0, 1) K }[/math] will not change the resulting Minkowski functional. The same is true of [math]\displaystyle{ L := (0, 1] K }[/math] and of [math]\displaystyle{ L := p_K^{-1}([0, 1]). }[/math]
    • If [math]\displaystyle{ D ~\stackrel{\scriptscriptstyle\text{def}}{=}~ \left\{y \in X : p_K(y) = 1 \text{ or } p_K(y) = 0 \right\} }[/math] then [math]\displaystyle{ p_D = p_K }[/math] and [math]\displaystyle{ D }[/math] has the particularly nice property that if [math]\displaystyle{ r \gt 0 }[/math] is real then [math]\displaystyle{ x \in r D }[/math] if and only if [math]\displaystyle{ p_D(x) = r }[/math] or [math]\displaystyle{ p_D(x) = 0. }[/math][note 1] Moreover, if [math]\displaystyle{ r \gt 0 }[/math] is real then [math]\displaystyle{ p_D(x) \leq r }[/math] if and only if [math]\displaystyle{ x \in (0, r] D. }[/math]
  5. Subadditive/Triangle inequality: [math]\displaystyle{ p_K }[/math] is subadditive if and only if [math]\displaystyle{ (0, 1) K }[/math] is convex. If [math]\displaystyle{ K }[/math] is convex then so are both [math]\displaystyle{ (0, 1) K }[/math] and [math]\displaystyle{ (0, 1] K }[/math] and moreover, [math]\displaystyle{ p_K }[/math] is subadditive.
  6. Scaling the set: If [math]\displaystyle{ s \neq 0 }[/math] is a scalar then [math]\displaystyle{ p_{s K}(y) = p_K\left(\tfrac{1}{s} y\right) }[/math] for all [math]\displaystyle{ y \in X. }[/math] Thus if [math]\displaystyle{ 0 \lt r \lt \infty }[/math] is real then [math]\displaystyle{ p_{r K}(y) = p_K\left(\tfrac{1}{r} y\right) = \tfrac{1}{r} p_K(y). }[/math]
  7. Symmetric: [math]\displaystyle{ p_K }[/math] is symmetric (meaning that [math]\displaystyle{ p_K(- y) = p_K(y) }[/math] for all [math]\displaystyle{ y \in X }[/math]) if and only if [math]\displaystyle{ (0, 1) K }[/math] is a symmetric set (meaning that[math]\displaystyle{ (0, 1) K = - (0, 1) K }[/math]), which happens if and only if [math]\displaystyle{ p_K = p_{- K}. }[/math]
  8. Absolute homogeneity: [math]\displaystyle{ p_K(u x) = p_K(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all unit length scalars [math]\displaystyle{ u }[/math][note 2] if and only if [math]\displaystyle{ (0, 1) u K \subseteq (0, 1) K }[/math] for all unit length scalars [math]\displaystyle{ u, }[/math] in which case [math]\displaystyle{ p_K(s x) = |s| p_K(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all non-zero scalars [math]\displaystyle{ s \neq 0. }[/math] If in addition [math]\displaystyle{ p_K }[/math] is also real-valued then this holds for all scalars [math]\displaystyle{ s }[/math] (that is, [math]\displaystyle{ p_K }[/math] is absolutely homogeneous[note 3]).
    • [math]\displaystyle{ (0, 1) u K \subseteq (0, 1) K }[/math] for all unit length [math]\displaystyle{ u }[/math] if and only if [math]\displaystyle{ (0, 1) u K = (0, 1) K }[/math] for all unit length [math]\displaystyle{ u. }[/math]
    • [math]\displaystyle{ s K \subseteq K }[/math] for all unit scalars [math]\displaystyle{ s }[/math] if and only if [math]\displaystyle{ s K = K }[/math] for all unit scalars [math]\displaystyle{ s; }[/math] if this is the case then [math]\displaystyle{ (0, 1) K = (0, 1) s K }[/math] for all unit scalars [math]\displaystyle{ s. }[/math]
    • The Minkowski functional of any balanced set is a balanced function.[8]
  9. Absorbing: If [math]\displaystyle{ K }[/math] is convex or balanced and if [math]\displaystyle{ (0, \infty) K = X }[/math] then [math]\displaystyle{ K }[/math] is absorbing in [math]\displaystyle{ X. }[/math]
    • If a set [math]\displaystyle{ A }[/math] is absorbing in [math]\displaystyle{ X }[/math] and [math]\displaystyle{ A \subseteq K }[/math] then [math]\displaystyle{ K }[/math] is absorbing in [math]\displaystyle{ X. }[/math]
    • If [math]\displaystyle{ K }[/math] is convex and [math]\displaystyle{ 0 \in K }[/math] then [math]\displaystyle{ [0, 1] K = K, }[/math] in which case [math]\displaystyle{ (0, 1) K \subseteq K. }[/math]
  10. Restriction to a vector subspace: If [math]\displaystyle{ S }[/math] is a vector subspace of [math]\displaystyle{ X }[/math] and if [math]\displaystyle{ p_{K \cap S} : S \to [0, \infty] }[/math] denotes the Minkowski functional of [math]\displaystyle{ K \cap S }[/math] on [math]\displaystyle{ S, }[/math] then [math]\displaystyle{ p_K\big\vert_S = p_{K \cap S}, }[/math] where [math]\displaystyle{ p_K\big\vert_S }[/math] denotes the restriction of [math]\displaystyle{ p_K }[/math] to [math]\displaystyle{ S. }[/math]
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 [math]\displaystyle{ A \subseteq X }[/math] that satisfies [math]\displaystyle{ (0, \infty) A = X }[/math] is necessarily absorbing in [math]\displaystyle{ X }[/math] is straightforward and can be found in the article on absorbing sets.

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

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

Examples

  1. If [math]\displaystyle{ \mathcal{L} }[/math] is a non-empty collection of subsets of [math]\displaystyle{ X }[/math] then [math]\displaystyle{ p_{\cup \mathcal{L}}(x) = \inf \left\{p_L(x) : L \in \mathcal{L} \right\} }[/math] for all [math]\displaystyle{ x \in X, }[/math] where [math]\displaystyle{ \cup \mathcal{L} ~\stackrel{\scriptscriptstyle\text{def}}{=}~ {\textstyle\bigcup\limits_{L \in \mathcal{L}}} L. }[/math]
    • Thus [math]\displaystyle{ p_{K \cup L}(x) = \min \left\{p_K(x), p_L(x) \right\} }[/math] for all [math]\displaystyle{ x \in X. }[/math]
  2. If [math]\displaystyle{ \mathcal{L} }[/math] is a non-empty collection of subsets of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ I \subseteq X }[/math] satisfies [math]\displaystyle{ \left\{x \in X : p_L(x) \lt 1 \text{ for all } L \in \mathcal{L}\right\} \quad \subseteq \quad I \quad \subseteq \quad \left\{x \in X : p_L(x) \leq 1 \text{ for all } L \in \mathcal{L}\right\} }[/math] then [math]\displaystyle{ p_I(x) = \sup \left\{p_L(x) : L \in \mathcal{L}\right\} }[/math] for all [math]\displaystyle{ x \in X. }[/math]

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

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

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

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

Positive homogeneity characterizes Minkowski functionals

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

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

  1. Strict positive homogeneity: [math]\displaystyle{ \; f(t x) = t f(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all positive real [math]\displaystyle{ t \gt 0. }[/math]
    • This statement is equivalent to: [math]\displaystyle{ f(t x) \leq t f(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all positive real [math]\displaystyle{ t \gt 0. }[/math]
  2. [math]\displaystyle{ f }[/math] is a Minkowski functional: meaning that there exists a subset [math]\displaystyle{ S \subseteq X }[/math] such that [math]\displaystyle{ f = p_S. }[/math]
  3. [math]\displaystyle{ f = p_K }[/math] where [math]\displaystyle{ K := \{x \in X : f(x) \leq 1\}. }[/math]
  4. [math]\displaystyle{ f = p_V \, }[/math] where [math]\displaystyle{ V \,:= \{x \in X : f(x) \lt 1\}. }[/math]

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

  1. Positive/Nonnegative homogeneity: [math]\displaystyle{ f(t x) = t f(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all nonnegative real [math]\displaystyle{ t \geq 0. }[/math]
Proof

If [math]\displaystyle{ f(t x) \leq t f(x) }[/math] holds for all [math]\displaystyle{ x \in X }[/math] and real [math]\displaystyle{ t \gt 0 }[/math] then [math]\displaystyle{ t f(x) = t f\left(\tfrac{1}{t}(t x)\right) \leq t \tfrac{1}{t} f(t x) = f(t x) \leq t f(x) }[/math] so that [math]\displaystyle{ t f(x) = f(t x). }[/math]

Only (1) implies (3) will be proven because afterwards, the rest of the theorem follows immediately from the basic properties of Minkowski functionals described earlier; properties that will henceforth be used without comment. So assume that [math]\displaystyle{ f : X \to [0, \infty] }[/math] is a function such that [math]\displaystyle{ f(t x) = t f(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all real [math]\displaystyle{ t \gt 0 }[/math] and let [math]\displaystyle{ K := \{y \in X : f(y) \leq 1\}. }[/math]

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

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

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

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

Characterizing Minkowski functionals on star sets

Proposition[10] — Let [math]\displaystyle{ f : X \to [0, \infty] }[/math] be any function and [math]\displaystyle{ K \subseteq X }[/math] be any subset. The following statements are equivalent:

  1. [math]\displaystyle{ f }[/math] is (strictly) positive homogeneous, [math]\displaystyle{ f(0) = 0, }[/math] and [math]\displaystyle{ \{x \in X : f(x) \lt 1\} \; \subseteq \; K \; \subseteq \; \{x \in X : f(x) \leq 1\}. }[/math]
  2. [math]\displaystyle{ f }[/math] is the Minkowski functional of [math]\displaystyle{ K }[/math] (that is, [math]\displaystyle{ f = p_K }[/math]), [math]\displaystyle{ K }[/math] contains the origin, and [math]\displaystyle{ K }[/math] is star-shaped at the origin.
    • The set [math]\displaystyle{ K }[/math] is star-shaped at the origin if and only if [math]\displaystyle{ t k \in K }[/math] whenever [math]\displaystyle{ k \in K }[/math] and [math]\displaystyle{ 0 \leq t \leq 1. }[/math] A set that is star-shaped at the origin is sometimes called a star set.[9]

Characterizing Minkowski functionals that are seminorms

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

Theorem — Let [math]\displaystyle{ K }[/math] be a subset of a real or complex vector space [math]\displaystyle{ X. }[/math] Then [math]\displaystyle{ p_K }[/math] is a seminorm on [math]\displaystyle{ X }[/math] if and only if all of the following conditions hold:

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

in which case [math]\displaystyle{ 0 \in K }[/math] and both [math]\displaystyle{ (0, 1) K = \{x \in X : p(x) \lt 1\} }[/math] and [math]\displaystyle{ \bigcap_{e \gt 0} (0, 1 + e) K = \left\{x \in X : p_K(x) \leq 1\right\} }[/math] will be convex, balanced, and absorbing subsets of [math]\displaystyle{ X. }[/math]

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

Corollary — If [math]\displaystyle{ K }[/math] is a convex, balanced, and absorbing subset of a real or complex vector space [math]\displaystyle{ X, }[/math] then [math]\displaystyle{ p_K }[/math] is a seminorm on [math]\displaystyle{ X. }[/math]

Positive sublinear functions and Minkowski functionals

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

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

Correspondence between open convex sets and positive continuous sublinear functions

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

Proof

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

See also

Notes

  1. ↑ It is in general false that [math]\displaystyle{ x \in r D }[/math] if and only if [math]\displaystyle{ p_D(x) = r }[/math] (for example, consider when [math]\displaystyle{ p_K }[/math] is a norm or a seminorm). The correct statement is: If [math]\displaystyle{ 0 \lt r \lt \infty }[/math] then [math]\displaystyle{ x \in r D }[/math] if and only if [math]\displaystyle{ p_D(x) = r }[/math] or [math]\displaystyle{ p_D(x) = 0. }[/math]
  2. ↑ [math]\displaystyle{ u }[/math] is having unit length means that [math]\displaystyle{ |u| = 1. }[/math]
  3. ↑ The map [math]\displaystyle{ p_K }[/math] is called absolutely homogeneous if [math]\displaystyle{ |s| p_K(x) }[/math] is well-defined and [math]\displaystyle{ p_K(s x) = |s| p_K(x) }[/math] for all [math]\displaystyle{ x \in X }[/math] and all scalars [math]\displaystyle{ s }[/math] (not just non-zero scalars).

References

  1. ↑ Narici & Beckenstein 2011, p. 109.
  2. ↑ Narici & Beckenstein 2011, p. 119.
  3. ↑ 3.0 3.1 Jarchow 1981, pp. 104-108.
  4. ↑ 4.0 4.1 Narici & Beckenstein 2011, pp. 115-154.
  5. ↑ 5.0 5.1 Schaefer 1999, p. 40.
  6. ↑ 6.0 6.1 6.2 Narici & Beckenstein 2011, p. 119-120.
  7. ↑ Kubrusly 2011, p. 200.
  8. ↑ 8.0 8.1 Schechter 1996, p. 316.
  9. ↑ Schechter 1996, p. 303.
  10. ↑ Schechter 1996, pp. 313-317.
  11. ↑ 11.0 11.1 Narici & Beckenstein 2011, pp. 192-193.

Further reading

  • F. Simeski, A.M.P. Boelens and M. Ihme. Modeling Adsorption in Silica Pores via Minkowski Functionals and Molecular Electrostatic Moments. Energies 13 (22) 5976 (2020). https://doi.org/10.3390/en13225976



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