Structure of slope-critical configurations

In 1970, P. R. Scott conjectured, and recently P. Ungar proved, that n noncollinear points in the plane determine at least n–1 directions. The goal here is to obtain information on the structure of the critical sets where the minimum of n–1 is attained. Such a set is distributed in a rather regular way onto spokes radiating out from a centrex point. If the number of spoke-pairs is even, the set must be centrally symmetric. Moreover, some bounds on the number of points per spoke are obtained. These results lead to the characterization, for small n, of certain types of slope-critical configurations.Dedicated to M. S. Irani on his 90th birthday 25 February, 1984Research supported in part by NSF EPSCoR Grant IPS-80-11451.