What is a Unitary Ball and How Do Different Metrics Shape Its Form?

[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|)

 

[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}

 

[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

a circle by radius 1?

 

[Bacle2]

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

 

[zendani]

thank you Balce2

i want it for a paper about friction

 

[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.

 

[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?

 

[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".