Counting Simplexes in R

A finite set of vectors S ⊆ R is called a simplex iff S is linearly dependent but all its proper subsets are independent. This concept arises in particular from stoichiometry. We are interested in this paper in the number of simplexes contained in some H ⊆ R , which we denote by simp(H). This investigation is particularly interesting for H spanning R and containing no collinear vectors. Our main result shows that for any H ⊆ R3 of fixed size not equal to 3, 4 or 7 and such that H spans R3 and contains no collinear vectors, simp(H) is minimal if and only if H is contained in two planes intersecting in H, and one of which is of size exactly 3. The minimal configurations for |H| = 3, 4, 7 are also completely described. The general problem for R remains open. ∗The research of the first author was partially supported by NSERC of Canada. †The research of the second author was partially supported by the Fund “Peregrinatio I” of MOL Co. Hungary, Grant no. 3/1994.