So Veit E.'s idea (as he now describes) involves what has been called "spherical polars" or "duality." I'll now re-do his solution so I can understand it (you too, maybe). That is, each new point on the sphere, defines an excluded hemiball (centered at it) in which the empty hemiball's center (if any) cannot lie. I.e. the sphere is cut by a new centered hyperplane every time you add new point. After N points added, we've cut the sphere using N "polar" hyperplanes into 2^N regions (some of these "regions" may not actually exist, i.e. are empty sets). Only the ++++...+++ region (if any) is eligible for occupancy by an empty-hemiball center (if any). The probability an empty hemiball exists, is the probability that this region exists. Now then Veit continues by arguing that by considering replacing points by their antipodes, all regions are "equivalent" i.e. under some assignment of {points, antipodes} any region you like becomes the +++...++++ region. Hence, if there are Reg(N,D) genuine regions, Veit argues the probability a +++...++++ region exists, is just Reg(N,D)/2^N. And Reg(N,D) is a combinatorial quantity that can be found using recurrences. In particular Reg(N,2) = 2*N obviously, explaining Salamin's formula. And Reg(N,3) = 2*(N-1) + Reg(N-1, 3) more generally Reg(N,D) = Reg(N-1, D-1) + Reg(N-1, D) by considering adding 1 new cut-hyperplane and considering the D-1 dimensional situation within that hyperplane. Hence Reg(N,D) always is a degree=D-1 polynomial in N which you can determine by fitting to the data for N=0,1,2,...,D-1. So the problem is now solved in every dimension. Very nice. Elegant.