Related papers: Algebraic Volume for Polytope Arise from Ehrhart T…

We overview the volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in $H^3$ and $S^3$. We also present some results, which provide a…

We show that the volume of the inner $r$-neighborhood of a polytope in the $d$-dimensional Euclidean space is a pluri-phase Steiner-like function, i.e. a continuous piecewise polynomial function of degree $d$, proving thus a conjecture of…

Preorder polytopes, defined from preorders on finite sets, are introduced and studied from a lattice point enumeration point of view. They naturally generalize arbor polytopes, recently introduced and studied by the second named author.…

We introduce a novel intrinsic volume concept in tropical geometry. This is achieved by developing the foundations of a tropical analog of lattice point counting in polytopes. We exhibit the basic properties and compare it to existing…

We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…

Cosmological polytopes of graphs are a geometric tool in physics to study wavefunctions for cosmological models whose Feynman diagram is given by the graph. After their recent introduction by Arkani-Hamed, Benincasa and Postnikov the focus…

Let $P$ be a bounded polyhedron defined as the intersection of the non-negative orthant ${\Bbb R}^n_+$ and an affine subspace of codimension $m$ in ${\Bbb R}^n$. We show that a simple and computationally efficient formula approximates the…

In a previous paper, we showed how to use the Ehrhart function $L_P(s)$, defined by $L_P(s) = \#(sP \cap \mathbb Z^d)$, to reconstruct a polytope $P$. More specifically, we showed that, for rational polytopes $P$ and $Q$, if $L_{P + w}(s) =…

We classify all the possible $delta$-vectors of d-dimensional integral convex polytopes whose volumes are less than or equal to 3/(d!).

A d-dimensional rational polytope P is a polytope whose vertices are located at the nodes of d-dimensional Z-lattice. Consider a number of points inside the inflated polytope (with coefficient of inflation k, k=1,2, 3...). The Ehrhart…

Classical results of Brent, Kuck and Maruyama (IEEE Trans. Computers 1973) and Brent (JACM 1974) show that any algebraic formula of size s can be converted to one of depth O(log s) with only a polynomial blow-up in size. In this paper, we…

We describe the implementation of algebraic polyhedra in Normaliz. In addition to convex hull computation/vertex enumeration, it is possible to compute triangulations, volumes, lattice points, face lattices and automorphism groups. The…

We prove a conjecture of Goncharov, which says that any multiple polylogarithm can be expressed via polylogarithms of depth at most half of the weight. We give an explicit formula for this presentation, involving a summation over trees that…

In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in…

We introduce a fast and robust algorithm for finding a plane $\Gamma$ with given normal $\vec{n}_\Gamma$, which truncates an arbitrary polyhedron $\mathcal{P}$ such that the remaining sub-polyhedron admits a given volume…

For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…

Ehrhart theory measures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, .... We compute the Ehrhart quasipolynomials of the standard Coxeter permutahedra for the classical Coxeter groups, expressing them…

We establish central limit theorems for natural volumes of random inscribed polytopes in projective Riemannian or Finsler geometries. In addition, normal approximation of dual volumes and the mean width of random polyhedral sets are…

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the…

The Lidskii formula for the type $A_n$ root system expresses the volume and Ehrhart polynomial of the flow polytope of the complete graph with nonnegative integer netflows in terms of Kostant partition functions. For every integer polytope…