# Unitary Ball

1. Mar 2, 2012

### 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|)

2. Mar 2, 2012

### quasar987

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}

3. Mar 2, 2012

### zendani

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

4. Mar 2, 2012

### Bacle2

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

5. Mar 2, 2012

### zendani

thank you Balce2

i want it for a paper about friction

6. Mar 3, 2012

### Matterwave

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.

7. Mar 3, 2012

### zendani

Thank you Matterwave

my answer for find unit ball is wrong?

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

8. Mar 3, 2012

### Matterwave

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".