Suppose that we choose \(n\) diameters in a sphere. Each diameter intersects the sphere's surface at  points, defining two hemispheres (one centered at each point.)

Now, a research paper I found (linked here: https://www.mathpages.com/home/kmath327/kmath327.htm) says:
"By choosing \(n\) diameters, we partition the surface into a certain number of subsets. Each of these subsets lies entirely inside or entirely outside the hemisphere centered on each of the \(n\) points."

I'm assuming that "subsets" refers to regions on the surface of the sphere...

But how does choosing \(n\) diameters partition the sphere into subsets? Can someone please describe this process intuitively?

Thank you so much!

 Dec 27, 2021

I tried comparing this to choosing \(n\) diameters for a circle. That clearly forms subsets of the circle (i.e. segments between the \(2n\) points of intersection between the diameters and the circumference)...


Maybe that can help answer the original question?

 Dec 28, 2021

Does anybody have an idea?

jsaddern  Dec 30, 2021

Maybe it consists of the regions where the induced hemispheres overlap?

 Dec 31, 2021

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