We study the convex hull PA of the 0-1 incidence vectors of all triangulations of a point configuration A. This was called the universal polytope in [4]. The affine span of PA is described in terms of the cocircuits of the oriented matroid of A. Its intersection with the positive orthant is a quasi-integral polytope QA whose integral hull equals PA . We present the smallest example where QA and PA differ. The duality theory for regular triangulations in [5] is extended to cover all triangulations. We discuss potential applications to enumeration and optimization problems regarding all triangulations.

TOPCOM is a package for computing triangulations of point configurations and oriented matroids. For example, for a point configuration one can compute the chirotope, components of the flip graph of triangulations, enumerate all triangulations. The core algorithms implemented in TOPCOM are described, and implentation issues are discussed.

In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum A 1 + \Delta \Delta \Delta +A r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding C (A 1 ; : : : ; A r ). In this paper we extend this correspondence in a natural way to cover also noncoherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos on subdivisions of Lawrence polytopes provides a new independent proof of the Bohne-Dress Theorem on zonotopal tilings. This application uses a combinatorial characterization of lifting subdivisions, also originally proved by Santos.

The theory of secondary and fiber polytopes implies that regular (also called convex or coherent) triangulations of configurations with n points in R

Abstract. We survey the generalized Baues problem of Billera and Sturmfels. The problem is one of discrete geometry and topology, and asks about the topology of the set of subdivisions of a certain kind of a convex polytope. Along with a discussion of most of the known results, we survey the motivation for the problem and its relation to triangulations, zonotopal tilings, monotone paths in linear programming, oriented matroid Grassmannians, singularities, and homotopy theory. Included are several open questions and problems. 1.

Abstract. We use the Cayley Trick to study polyhedral subdivisions of the product of two simplices. For arbitrary (fixed) l ≥ 2, we show that the numbers of regular and non-regular triangulations of ∆ l × ∆ k grow, respectively, as k Θ(k) and 2 Ω(k2). For the special case of ∆ 2 × ∆ k, we relate triangulations to certain class of lozenge tilings. This allows us to compute the exact number of triangulations up to k = 15, show that the number grows as e βk2 /2+o(k 2) where β ≃ 0.32309594 and prove that the set of all triangulations is connected under geometric bistellar flips. The latter has as a corollary that the toric Hilbert scheme of the determinantal ideal of 2×2 minors of a 3×k matrix is connected, for every k. We include “Cayley Trick pictures ” of all the triangulations of ∆ 2 × ∆ 2 and ∆ 2 × ∆ 3, as well as one non-regular triangulation of ∆ 2 × ∆ 5 and another of ∆ 3 × ∆ 3.

The cyclic polytope C(n; d) is the convex hull of any n points on the moment curve f(t; t 2 ; : : : ; t d ) : t 2 Rg in R d . For d 0 ? d, we consider the fiber polytope (in the sense of Billera and Sturmfels) associated to the natural projection of cyclic polytopes : C(n; d 0 ) ! C(n; d) which "forgets" the last d 0 \Gamma d coordinates. It is known that this fiber polytope has face lattice indexed by the coherent polytopal subdivisions of C(n; d) which are induced by the map . Our main result characterizes the triples (n; d; d 0 ) for which the fiber polytope is canonical in either of the following two senses: ffl all polytopal subdivisions induced by are coherent, ffl the structure of the fiber polytope does not depend upon the choice of points on the moment curve. We also discuss a new instance with a positive answer to the Generalized Baues Problem, namely that of a projection : P ! Q where Q has only regular subdivisions and P has two more vertices than its...

We are interested in a notion of elementary change between triangulations of a point configuration, the so-called bistellar flips, introduced by Gel'fand, Kapranov and Zelevinski. We construct sequences of triangulations of point configurations in dimension 3 with n 2 +2n+2 vertices and only 4n \Gamma 3 geometric bistellar flips (for every even integer n), and of point configurations in dimension 4 with arbitrarily many vertices and a bounded number of flips. This drastically improves previous examples and seems to be evidence against the conjecture that any two triangulations of a point configuration can be joined by a sequence of flips. Introduction Given a finite point configuration A in the Euclidean space R d of dimension d we call triangulations of A all the geometrically realized simplicial complexes which cover the convex hull of A and which have their sets of vertices contained in A. In this paper we are interested in a notion of vicinity or elementary change between tri...

We study the graph of bistellar flips between triangulations of a vector configuration A with d + 4 elements in rank d + 1 (i.e. with corank 3), as a step in the Baues problem. We prove that the graph is connected in general and 3-connected for acyclic vector configurations, which include all point configurations of dimension d with d + 4 elements. Hence, every pair of triangulations can be joined by a finite sequence of bistellar flips and every triangulation has at least 3 geometric bistellar neighbours. In corank 4, connectivity is not known and having at least 4 flips is false. In corank 2, the results are trivial since the graph is a cycle. Our methods are based in a dualization of the concept of triangulation of a point or vector configuration A to that of virtual chamber of its Gale transform B, introduced by de Loera et al. in 1996. As an additional result we prove a topological representation theorem for virtual chambers, stating that every virtual chamber of a rank 3 vector ...

We describe algorithms which solve two classical problems in lattice geometry for any fixed dimension d: the lattice covering and the simultaneous lattice packing–covering problem. Both algorithms involve semidefinite programming and are based on Voronoi’s reduction theory for positive definite quadratic forms which describes all possible Delone triangulations of Z d. Our implementations verify all known results in dimensions d ≤ 5. Beyond that we attain complete lists of all locally optimal solutions for d = 5. For d = 6 our computations produce new best known covering as well as packing–covering