A new polynomial-time algorithm for linear programming
Combinatorica
Constructing higher-dimensional convex hulls at logarithmic cost per face
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
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Mathematical Programming: Series A and B
A pivoting algorithm for convex hulls and vertex enumeration of arrangements and polyhedra
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
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Computational Geometry: Theory and Applications
The quickhull algorithm for convex hulls
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Computational Geometry: Theory and Applications
An Algorithm for Convex Polytopes
Journal of the ACM (JACM)
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Trees with convex faces and optimal angles
GD'06 Proceedings of the 14th international conference on Graph drawing
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We study the combinatorial complexity of D-dimensional polyhedra defined as the intersection of n halfspaces, with the property that the highest dimension of any bounded face is much smaller than D. We show that, if d is the maximum dimension of a bounded face, then the number of vertices of the polyhedron is O(nd) and the total number of bounded faces of the polyhedron is O(nd 2). For inputs in general position the number of bounded faces is O(nd). For any fixed 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 polynomial number of linear programs.