[1] Victor V. Batyrev and Benjamin Nill. Multiples of lattice polytopes without interior lattice points. Moscow Mathematical Journal 7:195–207, 2007. [2] Victor V. Batyrev, Benjamin Nill. Combinatorial aspects of mirror symmetry. Contemporary Mathematics, 452:35–66, 2008. [3] Barbara Baumeister, Christian Haase, Benjamin Nill and Andreas Paffenholz. On permutation polytopes. Advances in Mathema...

Convex polyhedra, i.e., the intersections of finitely many closed affine halfspaces inRd, are important objects in various areas of mathematics and other disciplines. In particular, the compact ones among them (polytopes), which equivalently can be defined as the convex hulls of finitely many points in Rd, have been studied since ancient times (e.g., the platonic solids). Polytopes appear as bu...

In the last years the vertex enumeration problem of polyhedra has seen a revival in the study of metabolic networks, which increased the demand for efficient vertex enumeration algorithms for high-dimensional polyhedra given by inequalities. In this paper we apply the concept of branch-decomposition to the vertex enumeration problem of polyhedra P = {x : Sx = b, x ≥ 0}. Therefore, we introduce ...

In the area of automatic parallelization of programs, restructura-tion of nested loops or parallelization of systems of aane recurrence equations use classically a geometrical model based on convex polyhedra of lattices. Since these algorithms are speciied for parametric sized calculation domains, para-metric lattice polyhedra have to be considered. Computing on such domains is a grand challeng...

Skeletal polyhedra and polygonal complexes in ordinary Euclidean 3-space are finite or infinite 3-periodic structures with interesting geometric, combinatorial and algebraic properties. They can be viewed as finite or infinite 3-periodic graphs (nets) equipped with additional structure imposed by the faces, allowed to be skew, zigzag or helical. A polyhedron or complex is regular if its geometr...

In the area of automatic parallelization of programs, restructura-tion of nested loops or parallelization of systems of aane recurrence equations use classically a geometrical model based on convex polyhedra of lattices. Since these algorithms are speciied for parametric sized calculation domains, para-metric lattice polyhedra have to be considered. Computing on such domains is a grand challeng...

To calculate the Minkowski-sum based similarity measure of two convex polyhedra, many relative orientations have to be considered. These relative orientations are characterized by the fact that some faces and edges of the polyhedra are parallel. For every relative orientation of the polyhedra, the volume or mixed volume of their Minkowski sum is evaluated. From the minimum of this volume, the s...

The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by “denting” at most two edges at a common vertex, and suspensions with a natu...

In this thesis, we define a static analysis by abstract interpretation of memory manipulations. It is based on a new numerical abstract domain, which is able to infer program invariants involving the operators min and max. This domain relies on tropical polyhedra, which are the analogues of convex polyhedra in tropical algebra. Tropical algebra refers to the set R∪{−∞} endowed with max as addit...