Minkowski functional

Key takeaways

Minkowski functional belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Minkowski functional to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Minkowski functional from memory before moving on to harder problems.

Reference excerpt

In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K {\textstyle K} is a subset of a real or complex vector space X , {\textstyle X,} then the Minkowski functional or gauge of K {\textstyle K} is defined to be the function p K : X → [ 0 , ∞ ] , {\textstyle p_{K}:X\to [0,\infty ],} valued in the extended real numbers, defined by

p K ( x ) = inf { r ∈ R : r > 0 and x ∈ r K } , x ∈ X , {\displaystyle p_{K}(x)=\inf\{r\in \mathbb {R} :r>0{\text{ and }}x\in rK\},\quad x\in X,}

where the infimum of the empty set is defined to be positive infinity. The set K {\textstyle K} is often assumed to have properties, such as being an absorbing disk in X {\textstyle X} , that guarantee that p K {\textstyle p_{K}} will be a seminorm on X . {\textstyle X.} In fact, every seminorm p {\textstyle p} on X {\textstyle X} is equal to the Minkowski functional (that is, p = p K {\textstyle p=p_{K}} ) of any subset K {\textstyle K} of X {\textstyle X} satisfying

{ x ∈ X : p ( x ) < 1 } ⊆ K ⊆ { x ∈ X : p ( x ) ≤ 1 } {\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 X {\textstyle X} and the first and last are also disks). Thus every seminorm (which is a function defined by purely algebraic properties) can be associated (non-uniquely) with an absorbing disk (which is a set with certain geometric properties) and conversely, every absorbing disk can be associated with its Minkowski functional (which will necessarily be a seminorm). These relationships between seminorms, Minkowski functionals, and absorbing disks is a major reason why Minkowski functionals are studied and used in functional analysis. In particular, through these relationships, Minkowski functionals allow one to "translate" certain geometric properties of a subset of X {\textstyle X} into certain algebraic properties of a function on X . {\textstyle X.}

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

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

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

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Minkowski functional

Start with the simplest possible case. Write down what Minkowski functional claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Minkowski functional before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Minkowski functional ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Minkowski functional

In research: Minkowski functional appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Minkowski functional in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Minkowski functional is common in secondary-school and first-year university syllabi. It links to neighbouring topics Convex analysis, Functional analysis, Hermann Minkowski, so understanding it makes those chapters shorter.
In everyday life: Look for Minkowski functional outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Minkowski functional in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Minkowski functional means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Minkowski functional out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Minkowski functional in simple terms?

In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K {\textstyle K} is a subset of a real or complex vector space X , {\textstyle X,} then the Minkowski functi…

Why does Minkowski functional matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Minkowski functional?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Minkowski functional.