Dimension 2, put 4 circles radius 1 at (1, 1), (-1, 1), (1, -1), (-1, -1). In the centre, put the largest circle you can. Then from the outside, you can't "touch" the inner circle.

Dimension 3, put 8 spheres radius 1 at (1, 1, 1), ... Then from the outside you can touch the inner sphere (and it's a bit bigger).

Once you get to a certain dimension (10 IIRC), the inner sphere is no longer in the convex hull of the outer spheres, it is "poking out" of the arrangement.

Spiky like that.

Oh they are very spike-y, well, my point in the above post is to just solve the eq but easier 'visualization' would be [0]

By the way, this is a similar phenomena to the 'curse of dimensionality' [1]

[0] https://www.youtube.com/watch?v=mceaM2_zQd8

[1] https://en.wikipedia.org/wiki/Curse_of_dimensionality

Most of the volume is near the edge of the sphere in higher dimensions. Closer to soap bubbles than what we consider to be true spheres.