The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings

Birkett Huber; Jörg Rambau; Francisco Santos

Displaying similar documents to “The Cayley Trick, lifting subdivisions and the Bohne-Dress theorem on zonotopal tilings”

Isocanted alcoved polytopes

María Jesús de la Puente, Pedro Luis Clavería (2020)

Applications of Mathematics

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Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$ -vectors and checking the validity of the following five conjectures: Bárány, unimodality, $3^{d}$ , flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$ , an isocanted alcoved polytope has $2^{d + 1} - 2$ vertices, its face lattice...

Volume thresholds for Gaussian and spherical random polytopes and their duals

Peter Pivovarov (2007)

Studia Mathematica

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Let g be a Gaussian random vector in ℝⁿ. Let N = N(n) be a positive integer and let $K_{N}$ be the convex hull of N independent copies of g. Fix R > 0 and consider the ratio of volumes $V_{N} : = v o l (K_{N} \cap R B ₂ ⁿ) / v o l (R B ₂ ⁿ)$ . For a large range of R = R(n), we establish a sharp threshold for N, above which $V_{N} \to 1$ as n → ∞, and below which $V_{N} \to 0$ as n → ∞. We also consider the case when $K_{N}$ is generated by independent random vectors distributed uniformly on the Euclidean sphere. In this case, similar threshold results are proved for both...

Delaunay polytopes derived from the Leech lattice

Mathieu Dutour Sikirić, Konstantin Rybnikov (2014)

Journal de Théorie des Nombres de Bordeaux

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A Delaunay polytope in a lattice $L$ is perfect if any affine transformation that preserve its Delaunay property is a composite of an homothety and an isometry. Perfect Delaunay polytopes are rare in low dimension and here we consider the ones that one can get in lattice that are sections of the Leech lattice. By doing so we are able to find lattices with several orbits of perfect Delaunay polytopes. Also we exhibit Delaunay polytopes which remain Delaunay in some superlattices....

Counting triangles that share their vertices with the unit $n$ -cube

Brandts, Jan, Cihangir, Apo

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This paper is about $0 / 1$ -triangles, which are the simplest nontrivial examples of $0 / 1$ -polytopes: convex hulls of a subset of vertices of the unit $n$ -cube $I^{n}$ . We consider the subclasses of right $0 / 1$ -triangles, and acute $0 / 1$ -triangles, which only have acute angles. They can be explicitly counted and enumerated, also modulo the symmetries of $I^{n}$ .

From binary cube triangulations to acute binary simplices

Brandts, Jan, van den Hooff, Jelle, Kuiper, Carlo, Steenkamp, Rik

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Cottle’s proof that the minimal number of $0 / 1$ -simplices needed to triangulate the unit $4$ -cube equals $16$ uses a modest amount of computer generated results. In this paper we remove the need for computer aid, using some lemmas that may be useful also in a broader context. One of the $0 / 1$ -simplices involved, the so-called antipodal simplex, has acute dihedral angles. We continue with the study of such acute binary simplices and point out their possible relation to the Hadamard determinant problem. ...

Approximation of the Euclidean ball by polytopes

Monika Ludwig, Carsten Schütt, Elisabeth Werner (2006)

Studia Mathematica

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There is a constant c such that for every n ∈ ℕ, there is an Nₙ so that for every N≥ Nₙ there is a polytope P in ℝⁿ with N vertices and $v o l ₙ (B ₂ ⁿ △ P) \leq c v o l ₙ (B ₂ ⁿ) N^{- 2 / (n - 1)}$ where B₂ⁿ denotes the Euclidean unit ball of dimension n.

On the $f$ - and $h$ -triangle of the barycentric subdivision of a simplicial complex

Sarfraz Ahmad (2013)

Czechoslovak Mathematical Journal

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For a simplicial complex $Δ$ we study the behavior of its $f$ - and $h$ -triangle under the action of barycentric subdivision. In particular we describe the $f$ - and $h$ -triangle of its barycentric subdivision $sd (Δ)$ . The same has been done for $f$ - and $h$ -vector of $sd (Δ)$ by F. Brenti, V. Welker (2008). As a consequence we show that if the entries of the $h$ -triangle of $Δ$ are nonnegative, then the entries of the $h$ -triangle of $sd (Δ)$ are also nonnegative. We conclude with a few properties of the $h$ -triangle of $sd (Δ)$ . ...

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne (2017)

Applications of Mathematics

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Acute triangles are defined by having all angles less than $π / 2$ , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension $n \geq 3$ , acuteness is defined by demanding that all dihedral angles between $(n - 1)$ -dimensional faces are smaller than $π / 2$ . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property...

An inequality concerning edges of minor weight in convex 3-polytopes

Igor Fabrici, Stanislav Jendrol' (1996)

Discussiones Mathematicae Graph Theory

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Let $e_{i j}$ be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is $20 e_{3, 3} + 25 e_{3, 4} + 16 e_{3, 5} + 10 e_{3, 6} + 6 [2 / 3] e_{3, 7} + 5 e_{3, 8} + 2 [1 / 2] e_{3, 9} + 2 e_{3, 10} + 16 [2 / 3] e_{4, 4} + 11 e_{4, 5} + 5 e_{4, 6} + 1 [2 / 3] e_{4, 7} + 5 [1 / 3] e_{5, 5} + 2 e_{5, 6} \geq 120$ ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

Complexity of the method of averaging

Dalík, Josef

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The general method of averaging for the superapproximation of an arbitrary partial derivative of a smooth function in a vertex $a$ of a simplicial triangulation $𝒯$ of a bounded polytopic domain in $ℜ^{d}$ for any $d \geq 2$ is described and its complexity is analysed.

On the combinatorial structure of $0 / 1$ -matrices representing nonobtuse simplices

Jan Brandts, Abdullah Cihangir (2019)

Applications of Mathematics

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A $0 / 1$ -simplex is the convex hull of $n + 1$ affinely independent vertices of the unit $n$ -cube $I^{n}$ . It is nonobtuse if none of its dihedral angles is obtuse, and acute if additionally none of them is right. Acute $0 / 1$ -simplices in $I^{n}$ can be represented by $0 / 1$ -matrices $P$ of size $n \times n$ whose Gramians $G = P^{⊤} P$ have an inverse that is strictly diagonally dominant, with negative off-diagonal entries. In this paper, we will prove that the positive part $D$ of the transposed inverse $P^{- ⊤}$ of $P$ is doubly stochastic and has the...

Computational results relating to problems concerning $σ (n)$

W. E. Mientka, R. L. Vogt (1970)

Matematički Vesnik

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Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

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A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space $_{φ}$ has pure discrete spectrum, we describe the collection $_{φ}^{P}$ of pairs of proximal tilings in $_{φ}$ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then $_{φ}$ and $_{ψ}$ are homeomorphic...