We consider a class A of generalized linear programs on the d-cube (due to Matoušek) and prove that Kalai’s subexponential simplex algorithm Random-Facet is polynomial on all actual linear programs in the class. In contrast, the subexponential analysis is known to be best possible for general instances in A. Thus, we identify a “geometric” property of linear programming that goes beyond all abs...

In this thesis we study various combinatorial problems relating to the geometry of point sets in the Euclidean plane. The unifying theme is that all the problems involve point sets that are not in general position, but have some collinearities. As well as giving rise to natural and interesting problems, the study of point sets with collinearities has important connections to other areas of math...

Combinatorial geometry is the study of order and incidence properties of groups of geometric features. Ordering properties for point sets in 2-D and 3-D can be seen as a generalization of ordering properties in 1-D and incidences are conngurations of features that are non-generic such as collinearity of points. By deening qualitative shape properties using combinatorial geometry we get a common...

In a previous paper (given at the International Congress of Mathematicians at Nice 1970) entitled "On the application of combinatorial analysis to number theory geometry and analysis", I discussed many combinatorial results and their applications . I will refer to this paper as I (as much as possible I will try to avoid overlap with this paper) . First I discuss those problems mentioned in I wh...