, The SCC of the dual set system induced by objects with union complexity ? is??is? is??(m) = O(?(m)/m), which together with Theorem 3 implies the stated bound. The value of ?(m) is O(m) for triangles with approximately same size [80], O(m log * m) for ?-fat triangles [48](where the constant of proportionality depends only on ?), and O(m2 log * m ) for locally ?-fat objects, Remember that the dual of a semi-algebraic set system is itself semi-algebraic (Lemma 1.3)

, R) be the set system induced on a set X of n points in R 2 by the family of axis-parallel rectangles. Aronov, Ezra, and Sharir [10] show that there is another set system R on X with SCC ? R (m, l) = O (l · log m)

, The other results follow from the facts that the SCC ?(m, l) ? of the primal set system induced on R 2 is O(m) for lines, O(ml) for strips, and O(ml 2 ) for cones, vol.86

, Macbeath Regions [1, +?). Perhaps some kind of interpolation could be defined on grid graphs of distinct dimensions, ? of the primal set system induced on R 2 by pseudodisks is

, ? Describe the exchange graphs for Terrain Guarding (see page 96). Can our lower bound construction be adapted to this problem?

, Prove the conjecture on page 97: the correct value of c 5 (the left-to-right ratio for planar 5-expanding graphs) is 3. More generally, determine c k for all k

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