Minkowski distance

Key takeaways

Minkowski distance 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 distance to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Minkowski distance from memory before moving on to harder problems.

Reference excerpt

The Minkowski distance or Minkowski metric is a family of distance functions on a real coordinate space, defined by taking the p {\displaystyle p} th power along each coordinate, given a positive parameter p {\displaystyle p} . That is, it is the distance function associated to a p-norm. It is a generalization of both the Euclidean distance ( p = 2 {\displaystyle p=2} ) and the Manhattan distance ( p = 1 {\displaystyle p=1} ). It is named after the mathematician Hermann Minkowski.

Definition The Minkowski distance of order p {\displaystyle p} (where p ≥ 1 {\displaystyle p\geq 1} ) between two points

X = ( x 1 , x 2 , … , x n ) and Y = ( y 1 , y 2 , … , y n ) ∈ R n {\displaystyle X=(x_{1},x_{2},\ldots ,x_{n}){\text{ and }}Y=(y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R} ^{n}}

is defined as:

D p ( X , Y ) = ( ∑ i = 1 n | x i − y i | p ) 1 p . {\displaystyle D_{p}\left(X,Y\right)={\biggl (}\sum _{i=1}^{n}|x_{i}-y_{i}|^{p}{\biggr )}^{\frac {1}{p}}.} This is the metric associated to the norm

‖ X ‖ p = ( ∑ i = 1 n | x i | p ) 1 / p {\displaystyle \|X\|_{p}=\left(\sum _{i=1}^{n}|x_{i}|^{p}\right)^{1/p}}

The Minkowski distance is a metric as a result of the Minkowski inequality,

‖ X + Y ‖ p ≤ ‖ X ‖ p + ‖ Y ‖ p . {\displaystyle \|X+Y\|_{p}\leq \|X\|_{p}+\|Y\|_{p}.}

Geometrically, this is a norm because the unit balls ‖ X ‖ p < 1 {\displaystyle \|X\|_{p}<1} are bounded and open in the standard topology, contain the origin, and are convex and balanced (invariant under X → − X {\displaystyle X\to -X} ). When 0 < p < 1 , {\displaystyle 0<p<1,} the same formula does not define a metric because it fails to satisfy the triangle inequality. For example, distance between ( 0 , 0 ) {\displaystyle (0,0)} and ( 1 , 1 ) {\displaystyle (1,1)} is 2 1 / p > 2 , {\displaystyle 2^{1/p}>2,} but the point ( 0 , 1 ) {\displaystyle (0,1)} is at a distance 1 {\displaystyle 1} from both of these points. However, a metric can be obtained for these values by removing the exponent of 1 / p {\displaystyle 1/p} :

d p ( X , Y ) = ∑ i = 1 n | x i − y i | p , 0 < p < 1. {\displaystyle d_{p}(X,Y)=\sum _{i=1}^{n}|x_{i}-y_{i}|^{p},\quad 0<p<1.}

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Minkowski distance

Start with the simplest possible case. Write down what Minkowski distance 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 distance 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 distance 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 distance

In research: Minkowski distance 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 distance 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 distance is common in secondary-school and first-year university syllabi. It links to neighbouring topics Distance, Hermann Minkowski, Metric geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Minkowski distance 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 distance in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Minkowski distance 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 distance out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Minkowski distance in simple terms?

The Minkowski distance or Minkowski metric is a family of distance functions on a real coordinate space, defined by taking the p {\displaystyle p} th power along each coordinate, given a positive parameter p {\displaystyle p} . That is, it is the distance function associated to a p-norm.

Why does Minkowski distance 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 distance?

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 distance.