Proof of Gru¨nbaum's conjecture on common transversals for translates
Discrete & Computational Geometry
Discrete & Computational Geometry
Stabbing pairwise disjoint translates in linear time
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Linear programming and convex hulls made easy
SCG '90 Proceedings of the sixth annual symposium on Computational geometry
The componentwise distance to the nearest singular matrix
SIAM Journal on Matrix Analysis and Applications
Finding a line transversal of axial objects in three dimensions
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
A subexponential bound for linear programming
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
A class of convex programs with applications to computational geometry
SCG '92 Proceedings of the eighth annual symposium on Computational geometry
On linear-time deterministic algorithms for optimization problems in fixed dimension
SODA '93 Proceedings of the fourth annual ACM-SIAM Symposium on Discrete algorithms
A Combinatorial Bound for Linear Programming and Related Problems
STACS '92 Proceedings of the 9th Annual Symposium on Theoretical Aspects of Computer Science
On geometric optimization with few violated constraints
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Bounded boxes, Hausdorff distance, and a new proof of an interesting Helly-type theorem
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Average case analysis of dynamic geometric optimization
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
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Recent combinatorial algorithms for linear programming also solve certain non-linear problems. We call these Generalized Linear Programming, or GLP, problems. One way in which convexity has been generalized by mathematicians is through a collection of results called the Helly theorems. We show that the every GLP problem implies a Helly theorem, and we give two paradigms for constructing a GLP problem from a Helly theorem. We give many applications, including linear expected time algorithms for finding line transversals and hyperplane fitting in convex metrics. These include GLP problems with the surprising property that the constraints are non-convex or even disconnected. We show that some Helly theorems cannot be turned into GLP problems.