Related papers: On the geometry of stack-sorting simplices

We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a…

The face lattice of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered…

The family of lattice simplices in $\mathbb{R}^n$ formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central role in problems in enumerative…

We show by a direct construction that there are at least $\exp\{cV^{(d-1)/(d+1)}\}$ convex lattice polytopes in $\mathbb{R}^d$ of volume $V$ that are different in the sense that none of them can be carried to an other one by a lattice…

We classify the three-dimensional lattice polytopes with two interior lattice points. Up to unimodular equivalence there are 22,673,449 such polytopes. This classification allows us to verify, for this case only, a conjectural upper bound…

We classify all tuples of lattice polyhedra of relative mixed volume 1 and all minimal (by inclusion) tuples of polyhedra of relative mixed volume 2. We also prove a conjecture by A. Esterov, which states that all tuples with finite…

We show that, if the interior of a lattice d-polytope P contains at least one lattice point, then it contains a lattice point whose coefficient of asymmetry with respect to P is at most b for some number b depending on d only. As an…

We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide…

Maximal mediated sets (MMS), introduced by Reznick, are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these…

It has been shown by Soprunov that the normalized mixed volume (minus one) of an $n$-tuple of $n$-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined…

We introduce Lipschitz functions on a finite partially ordered set $P$ and study the associated Lipschitz polytope $L(P)$. The geometry of $L(P)$ can be described in terms of descent-compatible permutations and permutation statistics that…

A theorem of Howe states that every 3-dimensional lattice polytope $P$ whose only lattice points are its vertices, is a Cayley polytope, i.e. $P$ is the convex hull of two lattice polygons with distance one. We want to generalize this…

We show that any point in the convex hull of each of (d+1) sets of (d+1) points in general position in \R^d is contained in at least (d+1)^2/2 simplices with one vertex from each set. This improves the known lower bounds for all d >= 4.

Consider a real point configuration $\mathbf{A}$ of size $n$ and an integer $r \leq n$. The vertices of the $r$-lineup polytope of $\mathbf{A}$ correspond to the possible orderings of the top $r$ points of the configuration obtained by…

We classify general systems of polynomial equations with a single solution, or, equivalently, collections of lattice polytopes of minimal positive mixed volume. As a byproduct, this classification provides an algorithm to evaluate the…

We prove that in any dimension $n$ there exists an origin-symmetric ellipsoid ${\mathcal{E}} \subset {\mathbb{R}}^n$ of volume $ c n^2 $ that contains no points of ${\mathbb{Z}}^n$ other than the origin, where $c > 0$ is a universal…

This note provides a simple proof for the equality between the normalized volume of a convex polytope with $m$ vertices and the mixed volume of $m$ simplices and thus shows the seemingly restrictive problem of computing mixed volume of…

Short and transparent proofs of central limit theorems for intrinsic volumes of random polytopes in smooth convex bodies are presented. They combine different tools such as estimates for floating bodies with Stein's method from probability…

We prove that every indefinite quadratic form with non-negative integer coefficients is the volume polynomial of a pair of lattice polygons. This solves the discrete version of the Heine-Shephard problem for two bodies in the plane. As an…

Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$.…