I am trying to understand L-infinity. The wiki definition is :
$\sup\limits_{n} |x_i|$. What does that say to me? (I am looking for a visual example).
Let's say I've got to points $A, B$ in 2d. I know the Euclidean distance is the line that connects them . What about the $L_{\infty}$?
I saw this picture in many resources:
I am not sure what it says . For example here I drew the Euclidean distance with blue, and by following the example of the previous photo I tried to see $L_{\infty}$ , but the red one looks like manhattan to me
:
If I try to apply the definition :
Let's say $A =(1, 3), B= (3, 2)$. Then I have $|x_A-x_B| = 2$ and $|y_A-y_B| = 1$ so is L-infinity : $\max(|x_A-x_B|, |y_A-y_B|)$?
I don't know what the phrase "I drew the Euclidean blue" means, I would have expected "I drew the Euclidean [insert noun here] blue", and I further would have expected the inserted noun to be "ball", but clearly you did not draw the Euclidean ball blue.
In that first picture, what is depicted is an $L_\infty$ ball centered at the origin $O=(0,0)$, meaning the set of all points $P=(x,y)$ such that $$\max\{|x-0|,|y-0|\} = \max\{|x|,|y|\} < r $$ (where the radius $r$ is undetermined in that picture).
The general intuition for all finite dimensional $L_\infty$ spaces is that $L_\infty$ balls are the same thing as Euclidean cubes: i.e. a square in $2$-dimensions; a true 3-dimensional cube in 3-dimensions; a 4-dimensional cube (a.k.a. a tesseract) in 4-dimensions; etc.