Abstract
We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement ofn line segments determined byh endpoints isO(h logh). While the previous upper bound,O(n?(n)), is tight for segments with distinct endpoints, it is far from being optimal whenn=Ω(h2). Our results show that, in a sense, the fundamental combinatorial complexity of a face arises not as a result of the number ofsegments, but rather as a result of the number ofendpoints.
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