Counting Simplexes in $R^3$

Abstract

A finite set of vectors ${\cal S} \subseteq {R}^n$ is called a simplex iff ${\cal S}$ 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 ${\cal H} \subseteq R^n$, which we denote by $simp({\cal H})$. This investigation is particularly interesting for ${\cal H}$ spanning $R^n$ and containing no collinear vectors.

Our main result shows that for any ${\cal H} \subseteq R^3$ of fixed size not equal to 3, 4 or 7 and such that ${\cal H}$ spans $R^3$ and contains no collinear vectors, $simp({\cal H})$ is minimal if and only if ${\cal H}$ is contained in two planes intersecting in ${\cal H}$, and one of which is of size exactly 3. The minimal configurations for $|{\cal H}|=3,4,7$ are also completely described.

The general problem for $R^n$ remains open.