Projective Equivalences of k-neighbourly Polytopes

Abstract

Let $2\le k\le \left\lfloor {\frac{d}{2}}\right\rfloor$ and let $\nu {(d, k)}$ be the largest number such that any set of $\nu {(d,k)}$ points lying in general position in $\mathbb {R}^d$ can be mapped by a permissible projective transformation onto the vertices of a k-neighborly polytope. The aim of this paper is to prove that $d + \left\lceil { \frac{d}{k}}\right\rceil +1 \le \nu {(d, k)} < 2d-k +1$.

Appendix: Oriented Matroids

This section summarizes in all detail every concept and result necessary for the full understanding of Sect. 3. It is intended as a fully referenced dictionary. The interested reader should consult [4, 8] for a full introduction to the theory of oriented matroids related to convexity. Here we will also cite some of the articles where results originally appeared.

Definition 11

(Oriented matroid) Let $E$ be a set (the ground set), a collection of signed sets $\mathcal {C}$ which are signed subsets of $E$ is the set of circuits of a matroid $\mathcal {M}$ if

• $\emptyset \not \in \mathcal {C}$,

• $\mathcal {C}=\mathcal {-C}$, where $\mathcal {-C}$ consists of inverting all the signs of each element on each circuit of $\mathcal {C}$,

• $\forall C, D \in \mathcal {C}$, if $\underline{C} \subset \underline{D}$, where $\underline{C}, \underline{D}$ stand for the sets $C$ and $D$ unsigned, then $C=D$ or $C=-D$,

• $\forall C, D \in \mathcal {C}$ such that $C \ne -D$ and $e \in C^+ \cap D^-$ there is an $A \in \mathcal {C}$ such that $A^+ \subset (C^+ \cup D^+) {\setminus } \{e \}$ and $A^- \subset (C^- \cup D^-) {\setminus } \{e \}$, where $C^{+}$ and $C^{-}$ denote the positive and negative elements of a circuit $C \in \mathcal {C}$.

Definition 12

(Oriented matroid of affine dependencies) Given a set of points $E$ in general position in $R^d$ we construct the oriented matroid of affine dependencies of $E$, $\mathcal {M}$, as follows:

• $E$ is its ground set;

• a subset $C \subset E$ is in the set of circuits of $\mathcal {M}$ iff $C$ is a minimal affine dependency;

• the positive and negative parts of $C$ are induced by the signs of the affine dependency.

It easily checked that this construction satisfies the oriented matroid circuit axioms. Observe that the signs associated to each of the elements in a circuit split those elements into a Radon partition.

A Radon partition of a set of points in general position encodes the dependencies between sets of $d+2$ points which, in its associated matroid, translates into circuits having exactly $d+2$ elements, that is, oriented matroids associated to configurations of points in general position have uniform rank.

Definition 13

(Rank) The rank in an oriented matroid associated to a configuration of points is the size of the maximum independent sets, in this case $d+1$.

Definition 14

(Bases) The bases of a matroid $\mathcal {M}$ are all maximal subsets of $E$ which contain no circuits.

In the case of a matroid of affine dependencies of points in general position, the set of basis of the matroid are all subsets of $E$ of cardinality equal to the rank of the matroid, $d+1$.

Definition 15

(Chirotope) A chirotope or basis orientation of an oriented matroid, $\mathcal {M}$, is a mapping $\chi$ of the set of ordered bases of $\mathcal {M}$ to $\{-1, 1\}$ satisfying;

• $\chi$ is alternating

• for any two ordered bases of $\mathcal {M}$ of the form $(e, e_1, \ldots e_r)$ and $(f, e_1, \ldots e_r)$, with$e \ne f$, we have $\chi (f, e_1, \ldots e_r)=-C(e)C(f)\chi (e, e_1, \ldots e_r)$, where $C$ is one of the two opposite signed circuits of $\mathcal {M}$ in the set $\{e, f, e_1, \ldots e_r \}$ and $C(e), C(f)$ are the signs of $e$ and $f$ in the circuit $C$, respectively.

An oriented matroid is said to be cyclic if it contains a circuit whose elements all have the same sign, otherwise $\mathcal {M}$ is said to be acyclic. It is easily proved that oriented matroids associated to affine dependencies of point configurations are always acyclic and uniform. [4, Sec.1.2].

Definition 16

(Reorientation) A reorientation of a matroid over the set $S \subset E$, is the process through which the sign of each element $s \in E$ in every circuit of in an oriented matroid $\mathcal {M}$ is reversed. This is, $\forall C \in \mathcal {M}$ we define ${}_{S} C$ as the circuit such that;

• ${}_{S} C^+= \{C^+ {\setminus } S\} \cup \{C^- \cap S\}$

• ${}_{S} C^-= \{C^- {\setminus } S\} \cup \{C^+ \cap S\}$

It is easy to check that the set of reoriented circuits of a matroid ${}_{-S}\mathcal {C}$ is indeed the set of circuits of a matroid which we will denote ${}_{S}\mathcal {M}$.

Theorem 2

(Las Vergnas [5]) Let $E$ be a finite set in $R^d$. Then there is an acyclic reorientation, ${}_{S}\mathcal {M}$, of its oriented matroid of affine dependencies $\mathcal {M}$, by reversing signs on $S \subset E$ iff there is a non-singular projective transformation, permissible for $E$, $T: R^d \rightarrow R^d$ such that ${}_{S}\mathcal {M}\equiv \mathcal {M}'$, where $\mathcal {M}'$ is the matroid of affine dependencies of $T(E)$ and the correspondence is given by the map $e \mapsto T(e)$.

For the definition of permissible projective transformation see Sect. 4.1, Definition 10.