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For $n=3$ this would just be a standard Venn diagram, because it would contain 8 different regions corresponding to the various combinations of intersections of sets the circles represent.

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  • $\begingroup$ What are your thoughts on the problem? And should the hyperspheres be unit spheres, or can their sizes vary? $\endgroup$ – Servaes May 17 at 6:15
  • $\begingroup$ Their sizes can vary. My only intuition is to place the centers of the hyperspheres on a regular $n-1$ simplex and hope a higher dimensional version of a venn diagram happens $\endgroup$ – cplusplusguru May 17 at 6:21
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Consider the $n-1$ coordinate hyperplanes, together with the unit (hyper-)sphere. Take a point not on any of these $n$ surfaces, and invert with respect to it. All the hyperplanes become spheres and the sphere remains a sphere.

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