Helly Theorems and Generalized Linear Programming

Recent combinatorial algorithms for linear programming can also be applied to certain non-linear problems. We call these Generalized Linear Programming, or GLP, problems. We connect this class to a collection of results from combinatorial geometry called Helly-type theorems. We show that there is a Helly-type theorem about the constrainst set of every GLP problem. Given a family H of sets with a Helly-type theorem, we give a paradigm for finding whether the intersection of H is empty, by formulating the question as a GLP problem. This leads to many applications, including linear expected time algorithms for finding line transversals and mini-max hyperplane fitting. Our applications include GLP problems with the suprising property that the constraints are non-convex or even disconnected