So while trying to improve my intuition about higher dimensions I found this video on youtube: Specific Part
Visualized for 2D(from the Video):
I would formulate the problem as follows:
d-cube $[-1,1]^d$ divided into $2^d$ d-cubes each containing a the largest centered ball
draw the d-ball centered thaat touches all unit d-balls
As said in the video the Volume V for the centered ball is $V \rightarrow \infty$ for $d \rightarrow \infty$
So I understand that the Volume of the d-balls in the unit d-cubes decreases and why and I can also proof that by calculating $\frac{Vball(d,r)}{Vcube(d,2r)} | d \rightarrow \infty$
Edit: And I also informally understand why the Volume of the centered d-ball increases
Could you hint me towards how a proof(sketch) could look like on how the radius of the centered ball behaves for: $d \rightarrow \infty$ ?