What are the different possibilities for the distance function in metric spaces?

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In summary, there are three common types of metrics for metric spaces on the Euclidean Plane: (A) with a distance function based on the circle equation, (B) with a distance function based on the maximum of coordinates, and (C) with a distance function based on the sum of coordinates. These correspond to a circle, square, and diamond shape, respectively, as shown in the attached picture. These are just a few examples of a larger class of metrics, including the p-metrics defined for every real p>=1. There are also many other types of metrics on R^2 besides these three.
  • #1
mynameisfunk
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OK, for metric spaces there are apparently 3 different possibilities for the distance function in M where M is the usual Euclidean Plane:

(A) D(u,v) = sqrt((x1-x2)2 + (y1-y2)2)
(B) D(u,v) = max(|x1-x2|,|y1-y2|)
(C) D(u,v) = |x1-x2| + |y1-y2|
which somehow correspond to the picture I have attached.
A corresponds to the circle, B to the square and C to the diamond(this is supposed to be a square diamond but i created the image in paint, sorry)
Now, I understand (A) but I cannot seem to understand why (B) and (C) end up looking this way. and to be honest, I don't understand B and C at all.
 

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  • #2
These are the unit balls with respect to each metric. In other words they mark the points which have the distance "1" to the origin (x_2, y_2) = (0,0). So the first one is a circle equation. The second one has the max of any coordinates, therefore max of (1,1) is 1 which is on the square. So figure out the diamond...

And you have definitely much more choices than 3. These are the most common three.
 
  • #3
To elaborate on trambolin's last sentence: these three are instances of a special class of metrics defined for every real p>=1:

[tex]d_p(x,y)=
\left(|x_1-y_1|^p+|x_2-y_2|^p\right)^{1/p}[/tex]

(A) corresponds to p=2
(C) corresponds to p=1
(B) is the extension for [itex]p=\infty[/itex]
Besides these p-metrics there are lots of other metrics on R^2.
 
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FAQ: What are the different possibilities for the distance function in metric spaces?

What is a metric space?

A metric space is a mathematical concept that defines the distance between any two points in a set. It is a generalization of the concept of distance in Euclidean space and can be used to measure the similarity or dissimilarity between objects.

What is the basic distance function in metric spaces?

The basic distance function in metric spaces is the metric or distance metric. It is a function that takes two points in a set and assigns a value representing the distance between them. The most common metric is the Euclidean distance, but there are other metrics such as Manhattan distance and Minkowski distance.

How is the distance between two points calculated in metric spaces?

The distance between two points in a metric space is calculated using the metric function. It takes into account the properties of the set and the chosen metric to determine the distance between the points. For example, in Euclidean space, the distance between two points (x1, y1) and (x2, y2) is calculated using the Pythagorean theorem: d = √[(x2-x1)^2 + (y2-y1)^2].

What are the properties of a metric space?

There are three main properties that a set must satisfy to be considered a metric space: the distance between any two points must be non-negative, the distance between a point and itself must be zero, and the distance between two points must be symmetric (i.e. the distance from point A to point B is the same as the distance from point B to point A). Additionally, the triangle inequality property must hold, which states that the distance between two points plus the distance between those points and a third point must be greater than or equal to the distance between the third point and the original two points.

How are metric spaces used in real-world applications?

Metric spaces have many practical applications, particularly in the fields of computer science, data analysis, and pattern recognition. They are used to measure the similarity or distance between objects, such as in image or text recognition, clustering algorithms, and recommendation systems. They are also used in optimization problems and in defining mathematical concepts such as fractals and topology.

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