# Can $n$ hyperspheres in $\mathbb{R}^{n-1}$ be placed so all $2^n$ partitions (in the Venn diagram sense) are realized?

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.

• What are your thoughts on the problem? And should the hyperspheres be unit spheres, or can their sizes vary? – Servaes May 17 at 6:15
• 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 – cplusplusguru May 17 at 6:21

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.