QUESTION (Dan Asimov): Suppose given N bi-infinite cylinders of radius 1 in 3-space, whose axes all contain the origin. What is the maximum number of (curved) faces that their common intersection can have? --I think the answer should be O(N) [so Grossman guess of a quadratic(N) is wrong]. Am I confused? More generally I think, if you intersect N convex 3-dimensional objects (to get a convex result of course) and the original objects' surfaces are defined by piecewise algebraic equations with bounded degree and bounded #pieces, then the final object should enjoy O(N) bounds on its face, edge, and vertex count, and can be computed via an O(NlogN)-time "randomized incremental" algorithm. Or at least this ought to be true of objects like spheres, halfspaces, or cylinders which have the property that when you intersect two, you get 1 convex object with at most two pieces discarded (if intersection nontrivial). The general idea to prove that is in: K.L.Clarkson & P.W.Shor: Applications of Random Sampling in Computational Geometry, II, Discrete and Computational Geometry 4,5 (1989) 387-421. and probably is also discussed in e.g. K.Mulmuley book.