New Lower Bounds for Convex Hull Problems in Odd Dimensions

We show that in the worst case, (ndd=2e;1 +n logn) sidedness queries are required to determine whether the convex hull of n points in IRd is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with (ndd=2e;1) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have (nbd=2c) facets, the previously best lower bound for these problems is only (n logn). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting a ne degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in IRd is dd=2esum-hard, in the sense of Gajentaan and Overmars. Key words. computational geometry, convex ...

• Type: Conference
• Year: 1996
• Where: COMPGEOM
• Authors: Jeff Erickson
• Keywords: COMPGEOM 1996, Convex Hull Facets, Convex Hulls, Discrete Geometry, Lower Bounds