# What is the correct definition of Minkowski distance

I realy confused. In a book ISBN: 978-0-470-27680-8 is written:

The Euclidean distance can be generalized as a special case of a family of metrics, called Minkowski distance or L p norm, defined as, $$D(\mathbf{x}_i,{\mathbf x}_j)=\left(\sum_{l=1}^d |x_{il}-x_{jl}|^{1/p}\right)^p \tag{1}$$

Is it correct? In the other sources Minkowski distance is defined as:

$$\left( \color{red}{\sum_{i=1}^n} |x_{i}-y_{i}|^p\right)^{1/p} \tag{2}$$

Which one is correct? (notice to powers)

J.D.'s edit: The highlighted red part was missing from a previous edit.. Originally, OP included the following two images:

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Please cite the book by its authors and title. ISBN numbers are not designed to replace bibliogrphic information and in fact suck at that! – Mariano Suárez-Alvarez Apr 5 '12 at 5:46
Might be that: Xu, Rui / Wunsch, Don, Clustering – draks ... Apr 5 '12 at 6:30
I think it's about similarity (1st formula) and dissimilarity (2nd formula) measures. – PHPst Apr 6 '12 at 4:51
– user2468 Apr 6 '12 at 4:56
Well, the wiki page on Minkowski_distance doesn't have references. – user2468 Apr 6 '12 at 5:01

A metric (or distance) and a norm are two different things. You can use a norm to define a metric, but not necessarily the other way around. The Minkowski metric is the metric induced by the $L_p$ norm, that is, the metric in which the distance between two vectors is the norm of their difference.

Both of these formulas describe the same family of metrics, since $p\to1/p$ transforms from one to the other. So if your question is "which of these two expressions describes the family of Minkowski metrics", the answer is both. However, if the question is "which of them gives the metric induced by the $L_p$ norm", it's the second one,

$$\left(\sum_{i=1}^n|x_i-y_i|^p\right)^{1/p}\;.$$

You can remember that by remembering that $L_2$ is the standard Euclidean norm, in which the exponent $2$ is on the inside and the exponent $1/2$ is on the outside.

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What is deference between norm and metric, in that book is mentioned:"Perhaps the most commonly used distance measure is the Euclidean distance, also known as L 2 norm, represented ..." but in euclidean distance D(0,4) for formula of book is equal to root(32) but for wikipedia formula it is equal to 16 – PHPst Apr 5 '12 at 10:45
@Reza: You'll have to point out what part of the answers you've already received you don't understand; otherwise there's no point in adding yet another one. Both Emre's answer and mine describe the difference between a norm and a metric as clearly as I can describe it; unless you say what part of that is unclear to you, there's nothing more I can say. Your book is simply wrong in conflating a distance and a norm. – joriki Apr 5 '12 at 10:49

If you are asking about the difference between a metric and a norm:

A norm and a metric are two related but different things. Generally speaking, a norm is a more "vector space" concept than metrics.

A norm assigns value to a SINGLE vectors (its length) in a vector space, while metric assigns value to TWO elements (their distance) in a metric space (which is not necessarily a vector space). Of course, in a vector space, a metric system can always be induced from a norm system by defining ||x-y|| as d(x,y).

Also, a norm must have Positive Homogeneity and Translation Invariance, both of which come quite naturally for a vector space, while a metric doesn't need to.

The Euclidean distance is the $L^2$ norm of the difference vector, and thus a special case as claimed. The first definition is correct; I can't read the second one.. See Wikipedia for a longer explanation.