Counting simplexes in \(\mathbb{R}^3\) (Q1130438)

The authors study the minimum number of simplices determined by a set of \(n\) vectors in dimension three which is full-dimensional and does not contain two collinear vectors. Here a simplex is just a minimal linearly dependent set, so the simplices need not be full-dimensional. Without the noncollinearity assumption, or if the set is planar, this problem is simple; for dimension three the extremal sets turn out to consist of \(n-2\) vectors in one plane, one of them coplanar with the remaining two vectors, with some exceptional configurations for \(n=3,4,7\). The minimal number of simplices then is \({n-2\choose 3} + {n-3\choose 2} +1\) for \(n\geq 4\). The authors also state some conjectures on the higher-dimensional analogues.