Bounds on the Complexity of Halfspace Intersections when the Bounded Faces have Small Dimension

For a polyhedron $$P$$P let $$B(P)$$B(P) denote the polytopal complex that is formed by all bounded faces of $$P$$P. If $$P$$P is the intersection of $$n$$n halfspaces in $$\mathbb R ^D$$RD, but the maximum dimension $$d$$d of any face in $$B(P)$$B(P) is much smaller, we show that the combinatorial complexity of $$P$$P cannot be too high; in particular, that it is independent of $$D$$D. We show that the number of vertices of $$P$$P is $$O(n^d)$$O(nd) and the total number of bounded faces of the polyhedron is $$O(n^{d^2})$$O(nd2). For inputs in general position the number of bounded faces is $$O(n^d)$$O(nd). We show that for certain specific values of $$d$$d and $$D$$D, our bounds are tight. For any fixed $$d$$d, we show how to compute the set of all vertices, how to determine the maximum dimension of a bounded face of the polyhedron, and how to compute the set of bounded faces in polynomial time, by solving a number of linear programs that is polynomial in $$n$$n.