Lower and upper bounds on the total weight of semi-rich acyclic arrangements of oriented lines in the plane (Q1567665)

The author defines the total weight of an arrangement of halfplanes (oriented lines) as the sum over all cells of the arrangement (vertices, edges and faces) of the number of halfplanes containing that cell. He then computes maximum and minimum of the total weight of an arrangement of \(n\) halfplanes in the very special class of arrangements he calls `semi-rich acyclic' arrangements. These are the arrangements of halfplanes in which there is a face that is contained in each halfplane, and which touches the bounding line of each halfplane. Thus it is essentially just the well-known cyclic arrangement (cyclic oriented matroid) with some additional tangential lines through the vertices of the central face. Thus the structure of the arrangement depends mainly on the numbers of additional lines through each vertex of that central face.