A b s t r ac t . The main purpose of this paper is to report on the state of the art of computing integer hulls and their facets as well as counting lattice points in convex polytopes. Using the polymake system we explore various algorithms and implementations. Our experience in this area is summarized in ten "rules of thumb". I n t ro d u c t i o nIn integer and linear optimization the software workhorses are solvers for linear programs (based on simplex or interior point methods) as well as generic frameworks for branch-andbound or branch-and-cut schemes. Comprehensive implementations are available both as Open Source, like SCIP [2], as well as commercial software, like CPLEX [24] and Gurobi [39]. While today it is common to solve linear programs with millions of rows and columns and, moreover, mixed integer linear programs with sometimes hundreds of thousands of rows and columns, big challenges remain. For instance, the 2010 version of the MIPLIB [30] lists the mixed-integer problem liu with 2 178 rows, 1 156 columns, and a total of only 10 626 non-zero coefficients; this seems to be impossible to solve with current techniques. One way to make progress in the field is to invent new families of cutting planes, either of a general kind or specifically tailored to a class of examples. In the latter situation the strongest possible cuts are obviously those arising from the facets of the (mixed) integer hull. A main purpose of this note is to report on the state of the art of getting at such facets in a brute force kind of way. And we will do so by explaining how our software system polymake [56] can help.Here we focus on integer linear programming (ILP); mixed integer linear programming (MILP) will only be mentioned in passing. To avoid technical ramifications we assume that all our linear programs (LP) are bounded. The brute force method for obtaining all facets of the integer hull is plain and simple, and it has two steps. First, we compute all the feasible integer points. Since we assumed boundedness these are only finitely many. Second, we compute the facets of their convex hull. Of course, the catch is that neither problem is really easy. Deciding if an ILP has an (integer) feasible point is known to be NP-complete [35]. So, we may not even hope for any efficient algorithm for the first step. Most likely, the situation for the second is about equally bad. While it is open whether or not there is a convex hull algorithm which runs in polynomial time measured in the combined sizes of the input and the output, recent work of Khachiyan et al.[47] indicates a negative answer. They show that computing the vertices of an unbounded polyhedron is hard; the difference to the general convex hull problem is that their result does not say anything about the rays of the polyhedron.Our paper is organized as follows. We start out with a very brief introduction to the polymake system and its usage. In Section 3 we explore how various convex hull algorithms and their implementations behave on various kinds of input. Our inpu...

Abstract. The d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d − 2 vertices are classified. In particular, it turns out that all of them are smooth Fano polytopes. This improves on previous results of (Casagrande, 2006) and (Øbro, 2008). Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.

A b s t r ac t . The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such that the vertex set of each facet forms a lattice basis. Casagrande showed that any smooth d-dimensional Fano polytope has at most 3d vertices. Smooth Fano polytopes in dimension d with at least 3d − 2 vertices are completely known. The main result of this paper deals with the case of 3d − k vertices for k fixed and d large. It implies that there is only a finite number of isomorphism classes of toric Fano d-folds X (for arbitrary d) with Picard number 2d − k such that X is not a product of a lower-dimensional toric Fano manifold and the projective plane blown up in three torus-invariant points. This verifies the qualitative part of a conjecture in a recent paper by the first author, Joswig, and Paffenholz.1. I n t ro d u c t i o n a n d m a i n r e s u lt s Let us first recall the basic definitions. We refer to [19,12] for more background. Let N ∼ = Z d be a lattice with associated real vector space N R := N ⊗ Z R isomorphic to R d . A polytope P is a convex, compact set in N R , its 0-dimensional faces are called vertices, and its faces of codimension 1 are called facets. If every facet F (of dimension d − 1) of a d-dimensional polytope P has exactly d vertices (i.e., F is a simplex), then P is called simplicial. The polytope P is called a lattice polytope if its vertices are lattice points (i.e., elements of N ).P is a lattice polytope, and P is full-dimensional and contains the origin 0 as an interior point, and for each facet F of P , the vertex set Vert F is a lattice basis of N . (ii) Two smooth Fano polytopes are lattice equivalent, if their vertex sets are in bijection by an affine-linear lattice automorphism.Remark 2. We decided to keep the notion of a smooth Fano polytope in order to be consistent with existing literature. However, we remark that there exists also the definition of a smooth polytope as a lattice polytope with unimodular vertex cones. A smooth Fano polytope is not a smooth polytope (but its dual polytope is).Note that any smooth Fano polytope P is necessarily simplicial. In each dimension there exist only finitely many smooth Fano polytopes up to lattice equivalence (we refer to the survey [19]). In 2007, Øbro described an explicit classification algorithm

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A b s t r ac t . The triangulations of point configurations which decompose as a free sum are classified in terms of the triangulations of the summands. The methods employ two new partially ordered sets associated with any triangulation of a point set with one marked point, the web of stars and the stabbing poset. Triangulations of smooth Fano polytopes are discussed as a case study.