Direction-Critical Configurations in Noncentral General Position

TL;DR

This paper proves Jamison's conjecture that large, centrally symmetric, direction-critical point configurations belong to four known families, extending the classification to noncentral general position and allowable sequences.

Contribution

It confirms Jamison's conjecture for centrally symmetric configurations in noncentral general position and classifies realizable allowable sequences.

Findings

Proves Jamison's conjecture for large configurations

Classifies centrally symmetric direction-critical configurations

Extends results to allowable sequences in noncentral general position

Abstract

In 1982, Ungar proved that the connecting lines of a set of $n$ noncollinear points in the plane determine at least $2\lfloor n/2 \rfloor$ directions (slopes). Sets achieving this minimum for $n$ odd (even) are called \emph{direction-(near)-critical} and their full classification is still open. To date, there are four known infinite families and over 100 sporadic critical configurations. Jamison conjectured that any direction-critical configuration with at least 50 points belongs to those four infinite families. Interestingly, except for a handful of sporadic configurations, all these configurations are centrally symmetric. We prove Jamison's conjecture, and extend it to the near-critical case, for centrally symmetric configurations in \emph{noncentral general position}, where only the connecting lines through the center of symmetry may pass through more than two points. As in Ungar's proof, our results are proved in the more general setting of \emph{allowable sequences}. We show that, up to equivalence, the \emph{central signature} of a set uniquely determines a centrally symmetric direction-(near)-critical allowable sequence in noncentral general position, and classify such allowable sequences that are geometrically realizable.