Depth contours in arrangements of halfplanes

Let H be a set of n halfplanes in R in general position, and let k < n be a given parameter. We show that the number of vertices of the arrangement of H that lie at depth exactly k (i.e., that are contained in the interiors of exactly k halfplanes of H) is O(nk + nk). The bound is tight when k = Θ(n). This generalizes the study of Dey [Dey98], concerning the complexity of a single level in an arrangement of lines, and coincides with it for k = O(n).