On the total weight of arrangements of halfplanes (Q1801629)

Consider a finite set of (open) halfplanes in the Euclidean plane, dividing the plane into convex cells. The weight of a cell is the number of halfplanes containing it. The total weight of the arrangement is the sum of the weights of all the cells. The author shows that for \(n\geq 3\), \(n\) odd, this total weight, \(W(H)\), attains a maximum if the intersection of the halfplanes is a regular \(n\)-gon, i.e., \(W(H)\leq n(3n^ 2+5)/8\) for odd \(n\). The method of proof appears not to apply in the case where \(n\) is even.