Combinatorial complexity bounds for arrangements of curves and spheres

We present upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike. For example, we prove that the maximum number of edges bounding m cells in an arrangement of n lines is Θ(m^2/3n^2/3 +n), and that it is O(m^2/3n^2/3β(n) +n) for n unit-circles, where β(n) (and later β(m, n)) is a function that depends on the inverse of Ackermann's function and grows extremely slowly. If we replace unit-circles by circles of arbitrary radii the upper bound goes up to O(m^3/5n^4/5β(n) +n). The same bounds (without the β(n)-terms) hold for the maximum sum of degrees of m vertices. In the case of vertex degrees in arrangements of lines and of unit-circles our bounds match previous results, but our proofs are considerably simpler than the previous ones. The maximum sum of degrees of m vertices in an arrangement of n spheres in three dimensions is O(m^4/7n^9/7β(m, n) +n²), in general, and O(m^3/4n^3/4β(m, n) +n) if no three spheres intersect in a common circle. The latter bound implies that the maximum number of unit-distances among m points in three dimensions is O(m^3/2β(m)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.