Günter M. Ziegler
We construct a 2-parameter family of 4-dimensional polytopes with extreme combinatorial structure: In this family, the ``fatness'' of the f-vector gets arbitrarily close to 9, the ``complexity'' (given by the flag vector) gets arbitrarily close to 16. The polytopes are obtained from suitable deformed products of even polygons by a projection to four-space.
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We introduce a deformed product construction for simple polytopes in terms of lower-triangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are ``strictly preserved'' under projection. Thus, starting from an arbitrary neighborly simplicial (d-2)-polytope Q on n-1 vertices we construct a deformed n-cube, whose projection to the last dcoordinates yields a neighborly cubical d-polytope. As an extension of thecubical case, we construct matrix representat…
Counting faces of randomly-projected polytopes when the projection radically lowers dimension
本文发展渐近方法计数随机高维多面体的面，并揭示在统计、概率、信息论和信号处理中的意外应用，包括高斯点云凸包、随机投影信号恢复以及高斯纠错码可纠正的粗差数量。
Pivot polytopes of products of simplices and shuffles of associahedra
本文提供了一个从单纯形积的轴心多胞体法扇到结合多面体洗牌法扇的分段线性同构。
These lecture notes treat some current aspects of two closely interrelated topics from the theory of convex polytopes: the shapes of f-vectors, and extremal constructions. The first lecture treats 3-dimensional polytopes; it includes a complete proof of the Koebe--Andreev--Thurston theorem, using the variational principle by Bobenko & Springborn (2004). In Lecture 2 we look at f-vector shapes of very high-dimensional polytopes. The third lecture explains a surprisingly simple construction for 2-simple 2-simplicial 4-polytopes, which have symmetric f-vectors. Lecture 4 sketches the geometry of …
An Intersection Product for the Polytope Algebra
本文通过多面体的交定义多面体代数的新乘法，研究其有限维子代数，并猜想及部分证明其满足硬Lefschetz性质与Hodge-Riemann关系。
We investigate necessary conditions for the existence of projections of polytopes that preserve full k-skeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to e-space strictly preserves the k-skeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of gr…
Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their k-skeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal and Ziegler's "projecting deformed products" construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that thi…
Indecomposability of 0/1-polytopes
本文证明了每个0/1多面体可以唯一地分解为不可分解多面体的Minkowski分解（考虑因子的平移），分解后的因子位于相互正交的子空间中。因此，每个0/1多面体都是不可分解0/1多面体的笛卡尔积。作为应用，我们得到了关于阶和链多面体、模形多面体、稳定集和团多面体、边多面体、流多面体以及2级/压缩多面体的统一组合不可分解性准则。我们还证明了每个非平凡的多项式因式分解都是不相交变量集合的多项式乘积。
We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal produc…
We construct a new 2-parameter family E_mn of self-dual 2-simple and 2-simplicial 4-polytopes, with flexible geometric realisations. E_44 is the 24-cell. For large m,n the f-vectors have ``fatness'' close to 6. The E_t-construction of Paffenholz and Ziegler applied to products of polygons yields cellular spheres with the combinatorial structure of E_mn. Here we prove polytopality of these spheres. More generally, we construct polytopal realisations for spheres obtained from the E_t-construction applied to products of polytopes in any dimension d>=3, if these polytopes satisfy some consistency …
We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially r…
In this paper we study $\times_0$-products of Lannér diagrams. We prove that every $\times_0$-product of at least four Lannér diagrams with at least one diagram of order $\ge 3$ is superhyperbolic. As a corollary, we obtain that known classifications exhaust all compact hyperbolic Coxeter polytopes that are combinatorially equivalent to products of simplices. We also consider compact hyperbolic Coxeter polytopes whose every Lannér subdiagram has order $2$. The second result of this paper slightly improves recent Burcroff's upper bound on the dimension of such polytopes to $12$.
Mahler's conjecture asks whether the cube is a minimizer for the volume product of a body and its polar in the class of symmetric convex bodies in a fixed dimension. It is known that every Hanner polytope has the same volume product as the cube or the cross-polytope. In this paper we prove that every Hanner polytope is a strict local minimizer for the volume product in the class of symmetric convex bodies endowed with the Banach-Mazur distance.
Although previous research has found several facts concerning chord lengths of regular polytopes, none of these investigations has considered whether any of these facts define relationships that might generalize to the chord lengths of all regular polytopes. Consequently, this paper explores whether four findings of previous studies-- viz., the four facts relating to the sums and products of squared chord lengths of regular polygons inscribed in unit circles-- can be generalized to all regular ($n$-dimensional) polytopes (inscribed in unit $n$-spheres). We show that (a) one of these four facts…
For a convex body $K \subset {\mathbb R}^n$, let $K^z = \{y\in{\mathbb R}^n : \langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of $K$ with respect to the center of polarity $z \in {\mathbb R}^n$. The goal of this paper is to study the maximum of the volume product $\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|$, among convex polytopes $K\subset {\mathbb R}^n$ with a number of vertices bounded by some fixed integer $m \ge n+1$. In particular, we prove that the supremum is reached at a simplicial polytope with exactly $m$ vertices and we provide a new proof of a res…
In this paper, we classify all the hyperbolic non-compact Coxeter polytopes of finite volume combinatorial type of which is either a pyramid over a product of two simplices or a product of two simplices of dimension greater than one. Combined with results of Kaplinskaja (1974) and Esselmann (1996) this completes the classification of hyperbolic Coxeter n-polytopes of finite volume with n+2 facets.
This thesis explores two specific topics of discrete geometry, the multitriangulations and the polytopal realizations of products, whose connection is the problem of finding polytopal realizations of a given combinatorial structure. A k-triangulation is a maximal set of chords of the convex n-gon such that no k+1 of them mutually cross. We propose a combinatorial and geometric study of multitriangulations based on their stars, which play the same role as triangles of triangulations. This study leads to interpret multitriangulations by duality as pseudoline arrangements with contact points cove…
A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $Ω= A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result i…
Gaussian polytopes with large Banach-Mazur distance to the cross-polytope
研究高斯多胞体与标准交叉多胞体的Banach-Mazur距离，证明当m=n^3时距离至少为c n^{5/8}(ln n)^{-5/8}的概率趋于1，改进了Friedland的指数4/7下界，方法结合离散化、条件化及K/U分解。
Random polytopes in convex bodies: Bridging the gap between extremal containers
本研究探讨块β分布族中随机点凸包构成的随机多面体的渐近性质，建立面期望数量的显式增长率，分析均匀分布下任意维度面的期望数量及体积差，揭示其插值性质以弥合两类凸容器中随机多面体行为的差距。
Many Facets in Random Polytopes from Product and Log-Concave Measures
该研究针对高维随机多面体的期望面数问题，推导了乘积测度下的下界、对数凹测度下的普适标度界，并构造非对数凹反例，揭示了测度性质对随机多面体面数的影响。
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