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Uniform decompositions of polytopes

Daniel Berend, Luba Bromberg (2006)

Applicationes Mathematicae

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 for the...

Unimodular Pisot substitutions and their associated tiles

Jörg M. Thuswaldner (2006)

Journal de Théorie des Nombres de Bordeaux

Let σ be a unimodular Pisot substitution over a d letter alphabet and let X 1 , ... , X d be the associated Rauzy fractals. In the present paper we want to investigate the boundaries ∂ X i ( 1 ≤ i ≤ d ) of these fractals. To this matter we define a certain graph, the so-called contact graph 𝒞 of σ . If σ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries ∂ X 1 , ... , ∂ X d . From this graph...

Uniqueness of Cartesian Products of Compact Convex Sets

Zbigniew Lipecki, Viktor Losert, Jiří Spurný (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

Let X i , i∈ I, and Y j , j∈ J, be compact convex sets whose sets of extreme points are affinely independent and let φ be an affine homeomorphism of ∏ i ∈ I X i onto ∏ j ∈ J Y j . We show that there exists a bijection b: I → J such that φ is the product of affine homeomorphisms of X i onto Y b ( i ) , i∈ I.

Universal container for packing rectangles

Janusz Januszewski (2002)

Colloquium Mathematicae

The aim of the paper is to find a rectangle with the least area into which each sequence of rectangles of sides not greater than 1 with total area 1 can be packed.

Universal tessellations.

David Singerman (1988)

Revista Matemática de la Universidad Complutense de Madrid

All maps of type (m,n) are covered by a universal map M(m,n) which lies on one of the three simply connected Riemann surfaces; in fact M(m,n) covers all maps of type (r,s) where r|m and s|n. In this paper we construct a tessellation M which is universal for all maps on all surfaces. We also consider the tessellation M(8,3) which covers all triangular maps. This coincides with the well-known Farey tessellation and we find many connections between M(8,3) and M.

Upper estimates on self-perimeters of unit circles for gauges

Horst Martini, Anatoliy Shcherba (2016)

Colloquium Mathematicae

Let M² denote a Minkowski plane, i.e., an affine plane whose metric is a gauge induced by a compact convex figure B which, as a unit circle of M², is not necessarily centered at the origin. Hence the self-perimeter of B has two values depending on the orientation of measuring it. We prove that this self-perimeter of B is bounded from above by the four-fold self-diameter of B. In addition, we derive a related non-trivial result on Minkowski planes whose unit circles are quadrangles.

Currently displaying 81 – 100 of 102