Abstract 
Recent combinatorial algorithms for linear programming can also be applied to certain nonlinear problems. We call these Generalized Linear Programming, or GLP, problems. We connect this class to a collection of results from combinatorial geometry called Hellytype theorems. We show that there is a Hellytype theorem about the constrainst set of every GLP problem. Given a family H of sets with a Hellytype 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 minimax hyperplane fitting. Our applications include GLP problems with the suprising property that the constraints are nonconvex or even disconnected.
