On the Complexity of Many Faces in Arrangements of Pseudo-Segments and of Circles

We obtain improved bounds on the complexity of m distinct faces in an arrangement of n pseudo-segments, n circles, or n unit circles. The bounds are worst-case optimal for unit circles; they are also worst-case optimal for the case of pseudo-segments, except when the number of faces is very small, in which case our upper bound is a polylogarithmic factor from the best-known lower bound. For general circles, the bounds nearly coincide with the best-known bounds for the number of incidences between m points and n circles, recently obtained in [9].