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I think this is a property spheres. It seems to me that any two spheres that are touching have a straight line from one center to the other center exactly through the point of contact. Try thinking of just two spheres and adding more in step-by-step.

Then the result follows because all the spheres are defined as centered on the cube/sub-cubes respectively.

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The inner sphere is not defined as centered on the cube; it is defined as touching all the other spheres.

That said, there is a symmetry argument that if it were centered anywhere else, something is wrong. But that only works if there is only one unique sphere that touches all the other spheres, which is also not obvious to me in higher dimensions.

You can go ahead and define it as centered on the cube. That still demonstrates the strange nature of high-dimensional spheres even if there wasn't a unique solution for touching all the other spheres.

Aha! Right, this is pretty convincing to me. Thanks!

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