Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results… (More)

This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the… (More)

The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n−d. That is, any two vertices of the polytope can be… (More)

We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 O(n−6) straight-edge triangulations of its convex hull. This improves the previous upper bound of… (More)

A pseudo-triangle is a simple polygon with exactly three convex vertices, and a pseudo-triangulation is a face-to-face tiling of a planar region into pseudo-triangles. Pseudo-triangulations appear as… (More)

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n− d.… (More)

In this paper we explicitly construct a triangulation of a 6-dimensional point configuration of 324 points which admits no geometric bistellar operations (or flips, for short). This triangulation is… (More)

Given an affine surjection of polytopes : P ! Q, the Generalized Baues Problem asks whether the poset of all proper polyhedral subdivisions of Q which are induced by the map has the homotopy type of… (More)