A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissecti...

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 connected by a path of at most n− d edges. This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope wit...

Motivated by recent work in parametric sequence alignment, we study the parameter space for scoring RNA folds and construct an RNA polytope. A vertex of this polytope corresponds to RNA secondary structures with common branching. We use this polytope and its normal fan to study the effect of varying three parameters in the free energy model that are not determined experimentally. Our results in...

This paper discusses ways of using a single panoramic image (captured by a rotating sensor-line camera having very-high spatial resolution) for the geometric shape recovery of a shown object. The objective is to create a sparse polyhedral model, only allowing a few interactive user inputs for a given single panoramic image. The study was motivated by the general question whether a single panora...

Automated building extraction from multi-source data has attracted great attention in recent years. This paper presents an approach to automated building extraction by fusion of image data, height data and 2D ground plans. In this approach buildings are detected using ground plans and height data. A split-and-merge process is applied to fuse image and height data and derive the parametric forms...

that contains the famous 19-face monostatic polyhedron polyhedra of this class allow to efficiently scan them , with a computer program, whether the class contains a .

The SQAP-polytope was associated to quadratic assignment problems with a certain symmetric objective function structure by Rijal (1995) and Padberg and Rijal (1996). We derive a technique for investigating the SQAP-polytope that is based on projecting the (low-dimensional) polytope into a lower dimensional vector-space, where the vertices have a “more convenient” coordinate structure. We exploi...

Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial) generalization of cyclic polytopes. We show that the colex order of facets of the ordinary polytope is a shelling order. This shelling shares many nice properties with the shellings of simplicial polytopes. We also give a shallow triangulation of the ordinary polytope, and show how the shelling and the triangulation are used...

The hyperdeterminant of format 2 × 2 × 2 × 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 Dequivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivision...

The order polytope O(P ) and the chain polytope C(P ) associated to a partially ordered set P are studied. In this paper, we introduce the convex polytope Γ(O(P ),−C(Q)) which is the convex hull of O(P ) ∪ (−C(Q)), where both P and Q are partially ordered sets with |P | = |Q| = d. It will be shown that Γ(O(P ),−C(Q)) is a normal and Gorenstein Fano polytope by using the theory of reverse lexico...