# Unit sphere intersected with hyperplanes, largest chamber?

I have a question which seems not that hard and maybe I can figure it out myself, but probably there is already a known theorem about it. Here it is:

Imagine the unit $n$-dimensional unit sphere is intersected by $n$ hyperplanes, passing through its center. These hyperplanes are slicing the sphere into $2^n$ parts. What is the easiest way to compare the sizes of these parts, to find which one is smallest/biggest? You can assume you are given the normal vectors to these hyperplanes and need to come up with a formula.

• You were mistaken to think this was not that hard! Jun 25, 2016 at 18:16

In two dimensions, those sectins are proportional to the angle between the enclosing lines. In three dimension the section is proportional to the enclosed solid angle. This can be generalized with the n-dimensional solid angle.

• Thank you for the response @flawr. For two dimensions it is indeed trivial. For three dimensions and higher though, I need several parameters for the computation. The solid angle formula calculates the area resulting from the intersection of the sphere with a cone (unless I missed something in the Wiki page). Jun 25, 2016 at 17:55
• I think the solid angle would be particularly useful if only bisected by a single hyperplane. Jun 25, 2016 at 18:07
• @RobertFrost But that is trivial, then you you just have two havles. Jun 25, 2016 at 18:11
• @ArturKirkoryan There are other formulae for polygons. Jun 25, 2016 at 18:12
• @flawr no, you would have an $n-1$ dimensional polytope. The question says $n$ hyperplanes. So in the case of a conventional sphere it's bisected by a pair of planes, with an unknown angle in-between and the peel of an obtuse orange-segment to measure the area of. The 3-sphere would have three planes, with I think six unknown angles. Jun 25, 2016 at 18:26