Delaunay polytopes derived from  the Leech lattice

Mathieu Dutour Sikirić; Konstantin Rybnikov

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Displaying similar documents to “Delaunay polytopes derived from the Leech lattice”

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...

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

Birkett Huber, Jörg Rambau, Francisco Santos (2000)

Journal of the European Mathematical Society

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In 1994, Sturmfels gave a polyhedral version of the Cayley Trick of elimination theory: he established an order-preserving bijection between the posets of coherent mixed subdivisions of a Minkowski sum $𝒜_{1} + \dots + 𝒜_{r}$ of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding $𝒞 (𝒜_{1}, \dots, 𝒜_{r})$ . In this paper we extend this correspondence in a natural way to cover also non-coherent subdivisions. As an application, we show that the Cayley Trick combined with results of Santos...

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.

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...

Uniform decompositions of polytopes

Daniel Berend, Luba Bromberg (2006)

Applicationes Mathematicae

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We design a method of decomposing convex polytopes into simpler polytopes. This decomposition yields a way of calculating exactly the volume of the polytope, or, more generally, multiple integrals over the polytope, which is equivalent to the way suggested in Schechter, based on Fourier-Motzkin elimination (Schrijver). Our method is applicable for finding uniform decompositions of certain natural families of polytopes. Moreover, this allows us to find algorithmically an analytic expression...

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.

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 $I^{n}$ .

Finite-to-one continuous s-covering mappings

Alexey Ostrovsky (2007)

Fundamenta Mathematicae

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The following theorem is proved. Let f: X → Y be a finite-to-one map such that the restriction $f | f^{- 1} (S)$ is an inductively perfect map for every countable compact set S ⊂ Y. Then Y is a countable union of closed subsets $Y_{i}$ such that every restriction $f | f^{- 1} (Y_{i})$ is an inductively perfect map.

Hyperreflexivity of bilattices

Kamila Kliś-Garlicka (2016)

Czechoslovak Mathematical Journal

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The notion of a bilattice was introduced by Shulman. A bilattice is a subspace analogue for a lattice. In this work the definition of hyperreflexivity for bilattices is given and studied. We give some general results concerning this notion. To a given lattice $ℒ$ we can construct the bilattice $Σ_{ℒ}$ . Similarly, having a bilattice $Σ$ we may consider the lattice $ℒ_{Σ}$ . In this paper we study the relationship between hyperreflexivity of subspace lattices and of their associated bilattices. Some examples...

Monotonically normal $e$ -separable spaces may not be perfect

John E. Porter (2018)

Commentationes Mathematicae Universitatis Carolinae

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A topological space $X$ is said to be $e$ -separable if $X$ has a $σ$ -closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that $e$ -separable PIGO spaces are perfect and asked if $e$ -separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of $e$ -separable monotonically normal spaces which are not perfect. Extremely normal $e$ -separable spaces are shown to be stratifiable.

Reducing the lengths of slim planar semimodular lattices without changing their congruence lattices

Gábor Czédli (2024)

Mathematica Bohemica

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Following G. Grätzer and E. Knapp (2007), a slim planar semimodular lattice, SPS lattice for short, is a finite planar semimodular lattice having no $M_{3}$ as a sublattice. An SPS lattice is a slim rectangular lattice if it has exactly two doubly irreducible elements and these two elements are complements of each other. A finite poset $P$ is said to be JConSPS-representable if there is an SPS lattice $L$ such that $P$ is isomorphic to the poset $J (Con L)$ of join-irreducible congruences of $L$ . We prove that...

Some methods to obtain t-norms and t-conorms on bounded lattices

Gül Deniz Çaylı (2019)

Kybernetika

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In this study, we introduce new methods for constructing t-norms and t-conorms on a bounded lattice $L$ based on a priori given t-norm acting on $[a, 1]$ and t-conorm acting on $[0, a]$ for an arbitrary element $a \in L ∖ {0, 1}$ . We provide an illustrative example to show that our construction methods differ from the known approaches and investigate the relationship between them. Furthermore, these methods are generalized by iteration to an ordinal sum construction for t-norms and t-conorms on a bounded lattice. ...

Sufficient conditions for a T-partial order obtained from triangular norms to be a lattice

Lifeng Li, Jianke Zhang, Chang Zhou (2019)

Kybernetika

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For a t-norm T on a bounded lattice $(L, \leq)$ , a partial order $\leq_{T}$ was recently defined and studied. In [11], it was pointed out that the binary relation $\leq_{T}$ is a partial order on $L$ , but $(L, \leq_{T})$ may not be a lattice in general. In this paper, several sufficient conditions under which $(L, \leq_{T})$ is a lattice are given, as an answer to an open problem posed by the authors of [11]. Furthermore, some examples of t-norms on $L$ such that $(L, \leq_{T})$ is a lattice are presented.

Less than $2^{ω}$ many translates of a compact nullset may cover the real line

Márton Elekes, Juris Steprāns (2004)

Fundamenta Mathematicae

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We answer a question of Darji and Keleti by proving that there exists a compact set C₀ ⊂ ℝ of measure zero such that for every perfect set P ⊂ ℝ there exists x ∈ ℝ such that (C₀+x) ∩ P is uncountable. Using this C₀ we answer a question of Gruenhage by showing that it is consistent with ZFC (as it follows e.g. from $c o f () < 2^{ω}$ ) that less than $2^{ω}$ many translates of a compact set of measure zero can cover ℝ.