# Unitary Ball: Exploring Metrics & Possibilities

• zendani
In summary, a unit ball is a ball of radius 1 in a given metric space. It is represented as B(x;1) and is defined as the set of all points y in the metric space that are within a distance of 1 from a given point x. The shape of a unit ball depends on the chosen distance function, with the most common one being a circular shape in 2-D and a spherical shape in 3-D. However, using alternative distance functions can result in different shapes such as squares or cubes.
zendani
I don't know anything about Unitary Ball

what is a Unitary ball? how make a unitary ball consider to a metric
(example: d(x,y) = max |xi-yi|)

Probably you mean "unit ball". This only means "a ball of radius 1". So, if (X,d) is a metric space, and x is a point in X, then the unit ball in (X,d) entered around x is the set

B(x;1) = {y in X | d(x,y)<1}

Thank you qusar987

so unit ball for d(x,y) = max |xi-yi| => B(x,1)= {y in x | maxi |xi-yi|<1}

maxi |xi-yi|<1 => |xi-yi|<1 => -1 <xi-yi<1 => 0<= xi-yi <1 => maxi (xi-yi)< 1

Why don't you try some simple examples, e.g., with xi=0, for xi real, then for xi in
R^2?

thank you Balce2

i want it for a paper about friction

If you use the max function as your distance function, then I think the "unit ball" is actually a square/cube/whatever you call one in higher dimensions.

Thank you Matterwave

my answer for find unit ball is wrong?

so i exactly can't recognize that unit ball will get which shape?

The usual distance function on R^n is d^2=sqrt(x^2+y^2+...), but that's not the only one you can use. You can certainly use your max function distance function.

The terminology "unit balls" comes from the usual distance function in which case, in 3-D you would get "balls". If you use other distance functions, you can get different shapes for your "unit balls".

## 1. What is a unitary ball?

A unitary ball is a mathematical concept that refers to a set of points in a multi-dimensional space that are all the same distance from the origin. It can also be thought of as a sphere with a radius of 1.

## 2. How is a unitary ball used in mathematics?

Unitary balls are commonly used in mathematics to explore different metrics and possibilities within a specific space. They can be used to define distances and measure properties of objects within the space.

## 3. What are some examples of metrics that can be explored using a unitary ball?

Some examples of metrics that can be explored using a unitary ball include Euclidean distance, Manhattan distance, and Minkowski distance. These metrics can be used to measure the distance between points in a space and can provide insights into the properties of the space.

## 4. How does the concept of a unitary ball relate to other mathematical concepts?

The concept of a unitary ball is closely related to other mathematical concepts such as normed vector spaces, metric spaces, and topology. These concepts all involve exploring and measuring properties within a space.

## 5. What are the real-world applications of studying unitary balls?

The study of unitary balls has many real-world applications in fields such as data analysis, computer science, and physics. It can be used to analyze and visualize complex data sets, optimize algorithms, and model physical systems.

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