This is a generalisation of this question, posted yesterday.
If we have a set of points $x_i$ on the surface of a sphere, or more generally a $d$-dimensional hypersphere, and we have a hyperspherical cap with a surface area a proportion $p$ of that of the whole sphere, is there always a position for the spherical cap such that it intersects no more than $p$ of the $x_i$?
The argument I used for the previous question doesn't seem to easily generalise because spherical caps can't cleanly tile a sphere.