我们希望提请注意两种理论之间有趣和有希望的相互作用。一方面，\textbf{pseudo-triangulations}理论对于隐式地解决卡彭特规则问题是有用的，并且后来被证明为图嵌入提供了一个很好的工具。另一方面，它是双曲虚多面体理论，它产生于一个古老的关于凸体的唯一性猜想（A. D. Alexandrov的问题：假设一个常数$C$在任何地方（非严格地）分离光滑三维凸体$K$的主曲率半径。那么$K$必然是一个半径为$C$的球。两个关键思想是：通过从平面伪三角形到球面伪平铺，我们避免了非点顶点。相反，我们使用伪二进制。给出了球面嵌入Laman加1图的一个定理。构造双曲多面体的困难问题可以归结为求球面嵌入图的问题。
英文摘要：
We wish to draw attention to an interesting and promising interaction of two theories.
On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved later to give a nice tool for graph embeddings.
On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov's problem): suppose that a constant $C$ separates (non-strictly) everywhere the principal curvature radii of a smooth 3-dimensional convex body $K$. Then $K$ is necessarily a ball of radius $C$.
The two key ideas are:
Passing from planar pseudo-triangulations to spherical pseudo-tilings, we avoid non-poited vertices. Instead, we use pseudo-di-gons. A theorem on spherically embedded Laman-plus-one graphs is announced.
The difficult problem of hyperbolic polytopes constructing can be reduced to finding spherically embedded graphs.
Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geometry, tropical oriented matroids, and to generalized permutohedra. We introduce a new approach to these objects, identifying a triangulation of a root polytope with a certain bijection between lattice points of two generalized permutohedra. In order to study such bijections, we define trianguloids as edge-colored graphs satisfying simple local axioms. We prove that trianguloids are in bijection with triangulations of r…
We characterize the polytopes in $\mathbb{R}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in $\mathbb{R}^d$ to be equidecomposable by lattice translations.
Given a polytope $\mathcal{P}$ in $\mathbb{R}^d$ and a subset $U$ of its vertices, is there a triangulation of $\mathcal{P}$ using $d$-simplices that all contain $U$? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of $\mathcal{P}$. Our proof relates triangulations of $\mathcal{P}$ to triangulations of its "shadow", a projection to a lower-dimensional space determined by $U$. In particular, we obtain a formula relating the volume of $\mathcal{P}$ with the volume of its shadow. This leads to an exact formula for the volume of a polytope …
Our main contribution is a concentration inequality for the symmetric volume difference of a $ C^2 $ convex body with positive Gaussian curvature and a circumscribed random polytope with a restricted number of facets, for any probability measure on the boundary with a positive density function. We also show that the Dirichlet-Voronoi tiling numbers satisfy $ \text{div}_{n-1} = (2πe)^{-1}(n+\ln n) + O(1)$, which improves a classical result of Zador by a factor of $o(n)$. In addition, we provide a remarkable open problem which is the natural geometric generalization of the famous and fundamental…
An $S$-hypersimplex for $S \subseteq \{0,1, \dots,d\}$ is the convex hull of all $0/1$-vectors of length $d$ with coordinate sum in $S$. These polytopes generalize the classical hypersimplices as well as cubes, crosspolytopes, and halfcubes. In this paper we study faces and dissections of $S$-hypersimplices. Moreover, we show that monotone path polytopes of $S$-hypersimplices yield all types of multipermutahedra. In analogy to cubes, we also show that the number of simplices in a pulling triangulation of a halfcube is independent of the pulling order.
To every poset P, Stanley (1986) associated two polytopes, the order polytope and the chain polytope, whose geometric properties reflect the combinatorial qualities of P. This construction allows for deep insights into combinatorics by way of geometry and vice versa. Malvenuto and Reutenauer (2011) introduced 'double posets', that is, (finite) sets equipped with two partial orders, as a generalization of Stanley's labelled posets. Many combinatorial constructions can be naturally phrased in terms of double posets. We introduce the 'double order polytope' and the 'double chain polytope' and we …
The circumcenter of mass of a simplicial polytope $P$ is defined as follows: triangulate $P$, assign to each simplex its circumcenter taken with weight equal to the volume of the simplex, and then find the center of mass of the resulting system of point masses. The so obtained point is independent of the triangulation. The aim of the present note is to give a definition of the circumcenter of mass that does not rely on a triangulation. To do so we investigate how volumes of polytopes change under Möbius transformations.
This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane. If a centrally symmetric octagon can form a lattice multiple tiling, then the multiplicity is at least seven. However, there are decagons which can form five-fold $($or six-fold$)$ lattice tilings. Consequently, whenever $n\ge 3$, there are non-parallelohedral polytopes which can form five-fold lattice tilings in the $n$-dimensional Euclidean space.
Tiling spaces are constructed using a metric in which two tilings of $\mathbb{R}^n$ are close if and only if, after a small translation, they agree on a large ball around the origin. We construct analogous spaces to study random triangulations of the plane. We construct a continuous space which is a foliated space equipped with a transverse measure, and a discrete space which is a transversal of that space. Measures on these triangulations can be constructed as limits of measures on spheres. We consider von Neumann algebras associated with these spaces. Under certain conditions, we show that t…
The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the "Simplicial Steinitz problem". It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the …
