Euclidean distance

This is taxicab geometry. You move from X to Y. Each square is 1.

Red:1+2+2+1+2+2=10

Green:5+5=10

They are the same line.

This is called Manhattan distance. n is the dimension.

Then, this is Euclidean norm.

This is 1*1 squares.

∴

There are 25 suares, so K=25. √50≒7.07.

50=5^2+5^2. This is the green.

Approximate Carathéodory’s Theorem

Higher dimensions are chaotic. However, you close to Zero, although you keep expanding.

This is almost 1-1=0. k go to ∞, which is called Euclidean norm.

Then you pile squares.

They are almost circles, and cubes are also balls. They are closed.

However, higher dimensions are opened, so they are chaotic.