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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 O(m2/~n 2/~ + n), and that it is O(m2/an2/3~(n) + n) for n unit-circles, where p(n) (and later/~(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(m3/Sn*/513(n) + n). The same bounds (without the B(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(m4/Tng/713(m, n) + n2), in general, and O(m3/4na/413(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 (m3/2fl(rn)) which improves the best previous upper bound on this problem. Applications of our results to other distance problems are also given.The techniques for studying this problem, both here and in [t9] and [20], can be viewed as divide-and-conquer attacks on this problem. In the approach taken in this paper, which we call the primal approach, the points are partitioned into groups according to an underlying subdivision of space into cells, to be specified shortly.We call this subdivision the funneling subdivision and its cells funnels, to distinguish 106

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We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets in d-dimensional space, d > 2, with linear storage and O(n ~) query time, d(d-1) c~= FT for all T>0. d(d-1)+l These bounds are better than those previously published for all d >-2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an e-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an e-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time. * D. Haussler gratefully acknowledges the support of ONR grant N00014-86-K-0454.

We consider the problem of computing geometric transformations (rotation, translation, reflexion) that map a point set A exactly or approximately into a point set B. We derive efficient algorithms for various cases (Euclidean or maximum metric, translation or rotation, or general congruence).