We study a combinatorial problem that recently arose in the context of shape optimization: among all triangles with vertices $(0,0)$, $(x,0)$, and $(0,y)$ and fixed area, which one encloses the most lattice points from $\mathbb{Z}_{{>}0}^{2}$? Moreover, does its shape necessarily converge to the isosceles triangle $(x=y)$ as the area becomes large? Laugesen and Liu suggested that, in contrast to similar problems, there might not be a limiting shape. We prove that the limiting set is indeed non-trivial and contains infinitely many elements. We also show that there exist “bad” areas where no triangle is particularly good at capturing lattice points and show that there exists an infinite set of slopes $y/x$ such that any associated triangle captures more lattice points than any other fixed triangle for infinitely many (and arbitrarily large) areas; this set of slopes is a fractal subset of $[1/3,3]$ and has Minkowski dimension of at most $3/4$.

For a non-empty polyhedral set $P\subset \mathbb{R}^{d}$, let ${\mathcal{F}}(P)$ denote the set of faces of $P$, and let $N(P,F)$ be the normal cone of $P$ at the non-empty face $F\in {\mathcal{F}}(P)$. We prove the identity

$$\begin{eqnarray}\mathop{\sum }_{F\in {\mathcal{F}}(P)}(-1)^{\operatorname{dim}F}\unicode[STIX]{x1D7D9}_{F-N(P,F)}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }P\text{ is bounded},\\ 0\quad & \text{if }P\text{ is unbounded and line-free}.\end{array}\right.\end{eqnarray}$$

$0$

We study a natural generalization of the noncrossing relation between pairs of elements in $[n]$ to $k$-tuples in $[n]$ that was first considered by Petersen et al. [J. Algebra324(5) (2010), 951–969]. We give an alternative approach to their result that the flag simplicial complex on $\binom{[n]}{k}$ induced by this relation is a regular, unimodular and flag triangulation of the order polytope of the poset given by the product $[k]\times [n-k]$ of two chains (also called Gelfand–Tsetlin polytope), and that it is the join of a simplex and a sphere (that is, it is a Gorenstein triangulation). We then observe that this already implies the existence of a flag simplicial polytope generalizing the dual associahedron, whose Stanley–Reisner ideal is an initial ideal of the Grassmann–Plücker ideal, while previous constructions of such a polytope did not guarantee flagness nor reduced to the dual associahedron for $k=2$. On our way we provide general results about order polytopes and their triangulations. We call the simplicial complex the noncrossing complex, and the polytope derived from it the dual Grassmann associahedron. We extend results of Petersen et al. [J. Algebra324(5) (2010), 951–969] showing that the noncrossing complex and the Grassmann associahedron naturally reflect the relations between Grassmannians with different parameters, in particular the isomorphism$G_{k,n}\cong G_{n-k,n}$. Moreover, our approach allows us to show that the adjacency graph of the noncrossing complex admits a natural acyclic orientation that allows us to define a Grassmann–Tamari order on maximal noncrossing families. Finally, we look at the precise relation of the noncrossing complex and the weak separability complex of Leclerc and Zelevinsky [Amer. Math. Soc. Transl.181(2) (1998), 85–108]; see also Scott [J. Algebra290(1) (2005), 204–220] among others. We show that the weak separability complex is not only a subcomplex of the noncrossing complex as noted by Petersen et al. [J. Algebra324(5) (2010), 951–969] but actually its cyclically invariant part.

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an $n$-point set $P$, and a collection ${\mathcal{F}}=\{F_{1},\ldots ,F_{m}\}$ of subsets of $P$, and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_{i}$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an $n$-point set in $d$-dimensional space, and ${\mathcal{F}}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$. Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well-known Tusnády problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author, Matoušek, and Talwar. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique $3$-sphere with the parameters $(f_{0},f_{1},f_{2},f_{3};f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex $4$-polytope. The $3$-sphere, which is $2$-simple and $2$-simplicial, was found by Werner [Linear constraints on face numbers of polytopes. PhD Thesis, TU Berlin, Germany, 2009]; we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: first, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

Even though there are various fast methods and preconditioning techniques available for the simulation of Poisson problems, little work has been done for solving Poisson's equation by using the Helmholtz decomposition scheme. To bridge this issue, we propose a novel efficient algorithm to solve Poisson's equation in irregular two dimensional domains for electrostatics through a quasi-Helmholtz decomposition technique—the loop-tree basis decomposition. It can handle Dirichlet, Neumann or mixed boundary problems in which the filling media can be homogeneous or inhomogeneous. A novel point of this method is to first find the electric flux efficiently by applying the loop-tree basis functions. Subsequently, the potential is obtained by finding the inverse of the gradient operator. Furthermore, treatments for both Dirichlet and Neumann boundary conditions are addressed. Finally, the validation and efficiency are illustrated by several numerical examples. Through these simulations, it is observed that the computational complexity of our proposed method almost scales as , where N is the triangle patch number of meshes. Consequently, this new algorithm is a feasible fast Poisson solver.

We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex $d$-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to $\unicode[STIX]{x1D70B}/2$.

There is always a natural embedding of $S_{s}\wr S_{k}$ into the linear programming (LP) relaxation permutation symmetry group of an orthogonal array integer linear programming (ILP) formulation with equality constraints. The point of this paper is to prove that in the $2$-level, strength-$1$ case the LP relaxation permutation symmetry group of this formulation is isomorphic to $S_{2}\wr S_{k}$ for all $k$, and in the $2$-level, strength-$2$ case it is isomorphic to $S_{2}^{k}\rtimes S_{k+1}$ for $k\geqslant 4$. The strength-$2$ result reveals previously unknown permutation symmetries that cannot be captured by the natural embedding of $S_{2}\wr S_{k}$. We also conjecture a complete characterization of the LP relaxation permutation symmetry group of the ILP formulation.

Supplementary materials are available with this article.

This paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also are applied to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. By well-known decompositions, it is sufficient to consider the case of affine cones $s+\mathfrak{c}$ , where $s$ is an arbitrary real vertex and $\mathfrak{c}$ is a rational polyhedral cone. For a given rational subspace  $L$ , we define the intermediate generating functions $S^{L}(s+\mathfrak{c})(\unicode[STIX]{x1D709})$ by integrating an exponential function over all lattice slices of the affine cone  $s+\mathfrak{c}$ parallel to the subspace  $L$ and summing up the integrals. We expose the bidegree structure in parameters $s$ and $\unicode[STIX]{x1D709}$ , which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math.12 (2012), 435–469] and [Intermediate sums on polyhedra: computation and real Ehrhart theory. Mathematika59 (2013), 1–22]. The bidegree structure is key to a new proof for the Baldoni–Berline–Vergne approximation theorem for discrete generating functions [Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. Contemp. Math.452 (2008), 15–33], using the Fourier analysis with respect to the parameter  $s$ and a continuity argument. Our study also enables a forthcoming paper, in which we study intermediate sums over multi-parameter families of polytopes.

A $(d-1)$-dimensional simplicial complex is called balanced if its underlying graph admits a proper $d$-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated lower bound theorem (LBT) for normal pseudomanifolds and characterize the case of equality; we introduce and characterize the balanced analog of the Walkup class; and we propose the balanced analog of the generalized lower bound conjecture (GLBC) and establish some related results. We close with constructions of balanced manifolds with few vertices.

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).

Supplementary materials are available with this article.

In previous work by Coates, Galkin and the authors, the notion of mutation between lattice polytopes was introduced. Such mutations give rise to a deformation between the corresponding toric varieties. In this paper we study one-step mutations that correspond to deformations between weighted projective planes, giving a complete characterization of such mutations in terms of T-singularities. We also show that the weights involved satisfy Diophantine equations, generalizing results of Hacking and Prokhorov.

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used, for instance, in integer linear programming.

We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of a combinatorial Gale transform, which is a duality between periodic difference equations of different orders. We describe periodic rational maps generalizing the classical Gauss map.

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.

Let ${\mathrm{OT} }_{d} (n)$ be the smallest integer $N$ such that every $N$-element point sequence in ${ \mathbb{R} }^{d} $ in general position contains an order-type homogeneous subset of size $n$, where a set is order-type homogeneous if all $(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in ${ \mathbb{R} }^{d} $ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in ${ \mathbb{R} }^{d} $. Two famous theorems of Erdős and Szekeres from 1935 imply that ${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} $. For $d\geq 3$, we give new bounds for ${\mathrm{OT} }_{d} (n)$. In particular, we show that ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } $, answering a question of Eliáš and Matoušek, and, for $d\geq 4$, we show that ${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height $d$ with $O(n)$ in the topmost exponent.

Let $d$ and $n$ be positive integers such that $n\geq d+ 1$ and ${\tau }_{1} , \ldots , {\tau }_{n} $ integers such that ${\tau }_{1} \lt \cdots \lt {\tau }_{n} $. Let ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )\subset { \mathbb{R} }^{d} $ denote the cyclic polytope of dimension $d$ with $n$ vertices $({\tau }_{1} , { \tau }_{1}^{2} , \ldots , { \tau }_{1}^{d} ), \ldots , ({\tau }_{n} , { \tau }_{n}^{2} , \ldots , { \tau }_{n}^{d} )$. We are interested in finding the smallest integer ${\gamma }_{d} $ such that if ${\tau }_{i+ 1} - {\tau }_{i} \geq {\gamma }_{d} $ for $1\leq i\lt n$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is normal. One of the known results is ${\gamma }_{d} \leq d(d+ 1)$. In the present paper a new inequality ${\gamma }_{d} \leq {d}^{2} - 1$ is proved. Moreover, it is shown that if $d\geq 4$ with ${\tau }_{3} - {\tau }_{2} = 1$, then ${C}_{d} ({\tau }_{1} , \ldots , {\tau }_{n} )$ is not very ample.