Thread: Question on the infinity metric

Hi guys,

We were learning about metric functions. I don't understand why d-infinity(0,1) is a square.

Similarly, why is a unit ball in R3 using the infinity metric a cube?

Can someone please explain this to me? Thanks very much.

I am afraid you will have to explain exactly what "d-infinity" you mean. The only "infinity metrics" I know are on function spaces ( ) and sequence spaces ( ). I don't know of any "d-infinity" on Euclidean spaces.

I am not sure if I am posting in the right section, but here it goes.

d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)

It is also called the supremum metric.

Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why.

Definition and examples of metric spaces

In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.

Can someone please explain why that is so?

Thanks very much.

Originally Posted by sakodo

I am not sure if I am posting in the right section, but here it goes. d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)
It is also called the supremum metric.
Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why. Definition and examples of metric spaces
In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.

First, the link does help. Now this completely standard material.
Second, What you first posted is misleading.
The set is a cube with center at the origin.
Here are some points in that set .

Originally Posted by sakodo

I am not sure if I am posting in the right section, but here it goes.

d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)

It is also called the supremum metric.

Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why.

Definition and examples of metric spaces

In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.

Can someone please explain why that is so?

Thanks very much.

Just a side note, sakodo: if you're unsure of where a thread belongs, report your own post (lower left corner, looks like a triangle with an exclamation mark inside), and a moderator will move it for you. I would recommend that this thread go to the Analysis subforum.

Originally Posted by sakodo

d-infinity is a distance function(a metric) defined as d(x,y)=max(|x1-y1|,|x2-y2|...)

It is also called the supremum metric.

Our lecturer told us that d-infinity(0,1) in R3 represents a cube, I had no idea why.

Definition and examples of metric spaces

In the above link, if you scroll down to example 5, it says d-infinity(0,1) in R2 is a square.

It doesn't actually say "d-infinity(0,1) in R2 is a square". It says that the unit circle is a square, for that metric d.

The unit circle is the set of points whose distance from the origin (as measured by the d-metric) is 1. If then . The equation says that either and , or and . If you think about what that means, it says that the point x lies on the square with vertices at .

Thanks for the reply guys. I think I kind of get it now.

So in R2, the unit circle of the d-metric is a set of points of which the maximum components are less than 1. This means the bounds for this metric are the lines x=1,-1 and y=1,-1. Thus the unit circle is a square. Is this correct? Same argument goes for R3.

But the unit ball B(0,1) , using the infinity metric, would be a square. I think I mixed the two up.

Thanks a lot guys.