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.

Simplexes and their applications - a short survey

“…In the proof we used the technique described in Definition 6 for reducing the dimension from H Â R 3 to H P R 2 , in the meanwhile we transformed linear algebraic simplexes into affine ones (see Definition1). The minimal configurations for jH j D 3; 4; 7 are almost completely described in [8], P.Sellers drew our attention to the missing one in a personal communication.…”

Minimum number of affine simplices of given dimension

“…The answer was given in [7]. Moreover, Problem 1 was generalized for matroids in [3]; actually its authors solved it a decade earlier than published, see [2].…”

“…A conjecture on both the minimum number and the structure attaining it is stated in [8]. The cases D = 3 and D = 4 were solved in [8] and [15], respectively.…”

“…In cases of low dimension, as solved in [8] and [15], almost all points of the extremal configurations for Problem 2 attaining the minimum number of affine simplexes lie on one or two lines, i.e. mostly contain affine simplexes of three points.…”

Counting Simplexes in $R^3$

“…In [1], we described which subsets of R n of fixed cardinality contain the largest or smallest number of simplexes, allowing collinear vectors. This problem relates to the potential maximal or minimal number of reactions in a given compound.…”