Radius of hypersphere for higher dimensions

Question

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: $d \rightarrow \infty$ ?

Answer 1

Ok the solution is too have a look at the main diagonal of the unit cubes. In 2D it is relatively easy to see that one edge of each square lies exactly in the center of the inscribed centered ball. Since the diameter of the unit ball is 1 we know that the radius of the central square is $\frac{{\sqrt d} -1 }{2} $ (Note that the maindiagonal of the d-cube of side length l is: ${\sqrt d}\ l$) since when we substract the diameter from the main diagonal, half of the rest is the radius. Since this obviously goes to infinity the radius for $d \rightarrow \infty$ is $\infty$