On the combinatorial complexity of euclidean Voronoi cells and convex hulls of d-dimensional spheres
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the nineteenth annual symposium on Computational geometry
Preprocessing chains for fast dihedral rotations is hard or even impossible
Computational Geometry: Theory and Applications
An optimal randomized algorithm for maximum Tukey depth
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
On the least median square problem
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
Lower bounds for linear degeneracy testing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Efficient detection of motion patterns in spatio-temporal data sets
Proceedings of the 12th annual ACM international workshop on Geographic information systems
Lower bounds for linear degeneracy testing
Journal of the ACM (JACM)
On FastMap and the Convex Hull of Multivariate Data: Toward Fast and Robust Dimension Reduction
IEEE Transactions on Pattern Analysis and Machine Intelligence
Counting and representing intersections among triangles in three dimensions
Computational Geometry: Theory and Applications
Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Construction of convex hull classifiers in high dimensions
Pattern Recognition Letters
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
The complexity of geometric problems in high dimension
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
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We show that in the worst case, $\Omega(n^{\ceil{d/2}-1} + n\log n)$ sidedness queries are required to determine whether the convex hull of n points in $\Real^d$ 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 $\Omega(n^{\ceil{d/2}-1})$ degenerate facets. While it has been known for several years that d-dimensional convex hulls can have $\Omega(n^{\floor{d/2}})$ facets, the previously best lower bound for these problems is only $\Omega(n\log n)$. Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in $\Real^d$ is $\ceil{d/2}$\SUM-hard in the sense of Gajentaan and Overmars.