Displaying similar documents to “Approximation of the Euclidean ball by polytopes”

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...

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....

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 + ⋯ + 𝒜 r of point configurations and of coherent polyhedral subdivisions of the associated Cayley embedding 𝒞 ( 𝒜 1 , ⋯ , 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...

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 ≥ 120 ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

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 ∩ 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 → 1 as n → ∞, and below which V N → 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...

Higher-dimensional cluster combinatorics and representation theory

Steffen Oppermann, Hugh Thomas (2012)

Journal of the European Mathematical Society

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Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate...

Gosset polytopes in integral octonions

Woo-Nyoung Chang, Jae-Hyouk Lee, Sung Hwan Lee, Young Jun Lee (2014)

Czechoslovak Mathematical Journal

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We study the integral quaternions and the integral octonions along the combinatorics of the 24 -cell, a uniform polytope with the symmetry D 4 , and the Gosset polytope 4 21 with the symmetry E 8 . We identify the set of the unit integral octonions or quaternions as a Gosset polytope 4 21 or a 24 -cell and describe the subsets of integral numbers having small length as certain combinations of unit integral numbers according to the E 8 or D 4 actions on the 4 21 or the 24 -cell, respectively. Moreover, we show...

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 .

On the ψ₂-behaviour of linear functionals on isotropic convex bodies

G. Paouris (2005)

Studia Mathematica

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The slicing problem can be reduced to the study of isotropic convex bodies K with d i a m ( K ) ≤ c √ n L K , where L K is the isotropic constant. We study the ψ₂-behaviour of linear functionals on this class of bodies. It is proved that | | ⟨ · , θ ⟩ | | ψ ₂ ≤ C L K for all θ in a subset U of S n - 1 with measure σ(U) ≥ 1 - exp(-c√n). However, there exist isotropic convex bodies K with uniformly bounded geometric distance from the Euclidean ball, such that m a x θ ∈ S n - 1 | | ⟨ · , θ ⟩ | | ψ ₂ ≥ c ∜ n L K . In a different direction, we show that good average ψ₂-behaviour of linear functionals...

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. ...

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

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Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation f = ∑ C ∈ a C φ C with a C ∈ E and a partition of unity φ C : C ∈ subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction f | P attains its extrema at vertices of P. Finally, a class of extremal functions on the...