Benjamin Nill
A reflexive polytope, respectively its associated Gorenstein toric Fano variety, is called pseudo-symmetric, if the polytope has a centrally symmetric pair of facets. Here we present a complete classification of pseudo-symmetric simplicial reflexive polytopes together with some applications. This generalizes a result of Ewald on pseudo-symmetric nonsingular toric Fano varieties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The reflexive dimension refldim(P) of a lattice polytope P is the minimal d so that P is the face of some d-dimensional reflexive polytope. We show that refldim(P) is finite for every P, and give bounds for refldim(kP) in terms of refldim(P) and k.
Lattices generated by lattice points in skeletons of reflexive polytopes are essential in determining the fundamental group and integral cohomology of Calabi-Yau hypersurfaces. Here we prove that the lattice generated by all lattice points in a reflexive polytope is already generated by lattice points in codimension two faces. This answers a question of J. Morgan.
We introduce reflexive polytopes of index l as a natural generalisation of the notion of a reflexive polytope of index 1. These l-reflexive polytopes also appear as dual pairs. In dimension two we show that they arise from reflexive polygons via a change of the underlying lattice. This allows us to efficiently classify all isomorphism classes of l-reflexive polygons up to index 200. As another application, we show that any reflexive polygon of arbitrary index satisfies the famous "number 12" property. This is a new, infinite class of lattice polygons possessing this property, and extends the p…
Recent work has focused on the roots z of the Ehrhart polynomial of a lattice polytope P. The case when Re(z) = -1/2 is of particular interest: these polytopes satisfy Golyshev's "canonical line hypothesis". We characterise such polytopes when dim(P) <= 7. We also consider the "half-strip condition", where all roots z satisfy -dim(P)/2 <= Re(z) <= dim(P)/2-1, and show that this holds for any reflexive polytope with dim(P) <= 5. We give an example of a 10-dimensional reflexive polytope which violates the half-strip condition, thus improving on an example by Ohsugi--Shibata in dimension 34.
It is known that every lattice polytope is unimodularly equivalent to a face of some reflexive polytope. A stronger question is to ask whether every $(0,1)$-polytope is unimodularly equivalent to a facet of some reflexive polytope. A large family of $(0,1)$-polytopes are the edge polytopes of finite simple graphs. In the present paper, it is shown that, by giving a new class of reflexive polytopes, each edge polytope is unimodularly equivalent to a facet of some reflexive polytope. Furthermore, we extend the characterization of normal edge polytopes to a characterization of normality for these…
Reflexive polytopes and the Picard ranks of Gorenstein toric Fano varieties
该研究推广了Eikelberg仿射依赖理论，证明维数不小于3的极化Gorenstein环面Fano簇对的Picard秩之和的上界，且该上界在极化对为单-单纯对时取到。
Let $δ(\mathcal{P})$ be the $δ$-vector of a reflexive polytope $\mathcal{P} \subset \mathbb{R}^d$ of dimension $d$ and $δ(\mathcal{P} ^\vee)$ the $δ$-vector of the dual polytope $\mathcal{P}^\vee \subset \mathbb{R}^d$. In general, $δ(\mathcal{P})=δ(\mathcal{P}^\vee)$ does not hold. In this paper, we give a higher-dimensional construction of reflexive polytope whose $δ$-vector equals the $δ$-vector of the dual polytope. In particular, we consider the case that the reflexive polytope and the dual polytope are unimodularly equivalent.
We classify terminal simplicial reflexive d-polytopes with 3d-1 vertices. They turn out to be smooth Fano d-polytopes. When d is even there is 1 such polytope up to isomorphism, while there are 2 when d is uneven.
The Ehrhart function $L_P(t)$ of a polytope $P$ is usually defined only for integer dilation arguments $t$. By allowing arbitrary real numbers as arguments we may also detect integer points entering (or leaving) the polytope in fractional dilations of $P$, thus giving more information about the polytope. Nevertheless, there are some polytopes that only gain new integer points for integer values of $t$; that is, these polytopes satisfy $L_P(t) = L_P(\lfloor t \rfloor)$. We call those polytopes semi-reflexive. In this paper, we give a characterization of these polytopes in terms of their hyperpl…
A classification of discrete polymatroids whose independence polytopes are reflexive will be presented.
Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on Grönbner bases, a new class of reflexive polytopes which possess the integer decomposition property and which arise from perfect graphs will be presented. Furthermore, the Ehrhart $δ$-polynomials of these polytopes will be studied.
One may construct a large class of Calabi-Yau varieties by taking anticanonical hypersurfaces in toric varieties obtained from reflexive polytopes. If the intersection of a reflexive polytope with a hyperplane through the origin yields a lower-dimensional reflexive polytope, then the corresponding Calabi-Yau varieties are fibered by lower-dimensional Calabi-Yau varieties. A top generalizes the idea of splitting a reflexive polytope into two pieces. In contrast to the classification of reflexive polytopes, there are infinite families of equivalence classes of tops. Tops may be used to describe …
A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. In this paper, we investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets
A new class of magic positive Ehrhart polynomials of reflexive polytopes
本文构造了一类新的具有魔正Ehrhart多项式的自反多面体，证明Stasheff多面体满足该性质，并部分证明圈的对称边多面体对偶的魔正性。
Reflexive polytopes which have the integer decomposition property are of interest. Recently, some large classes of reflexive polytopes with integer decomposition property coming from the order polytopes and the chain polytopes of finite partially ordered sets are known. In the present paper, we will generalize this result. In fact, by virtue of the algebraic technique on Gröbner bases, new classes of reflexive polytopes with the integer decomposition property coming from the order polytopes of finite partially ordered sets and the stable set polytopes of perfect graphs will be introduced. Furt…
Oda's conjecture for reflexive polytopes: some special cases
本文在若干特殊条件下证明了自反多胞体的Oda猜想，包括单纯自反多胞体与含原点的格多胞体满足特定顶点和格点条件的情形，以及面幺模多胞体和几乎余幺模自反多胞体对的情形。
Likelihood Geometry of Reflexive Polytopes
本文研究自反多胞形定义的统计模型的ML度，计算了所有4319类三维自反多胞形的ML度，发现与环簇度的间隙，并给出超立方体等族的ML度公式及ML度为1的嵌入。
It was observed by Bump et al. that Ehrhart polynomials in a special family exhibit properties similar to the Riemann ζ function. The construction was generalized by Matsui et al. to a larger family of reflexive polytopes coming from graphs. We prove several conjectures confirming when such polynomials have zeros on a certain line in the complex plane. Our main new method is to prove a stronger property called interlacing.
An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal $h^*$-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the $h^*$-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements.
Facet numbers of non-centrally symmetric reflexive polytopes arising from posets
本文研究了双链多面体的面数，证明其至多为6^{d/2}，并为Nill猜想提供了部分答案。
The correspondence between Gorenstein Fano toric varieties and reflexive polytopes has been generalized by Ilten and Süß to a correspondence between Gorenstein Fano complexity-one $T$-varieties and Fano divisorial polytopes. Motivated by the finiteness of reflexive polytopes in fixed dimension, we show that over a fixed base polytope, there are only finitely many Fano divisorial polytopes, up to equivalence. We classify two-dimensional Fano divisorial polytopes, recovering Huggenberger's classification of Gorenstein del Pezzo $\mathbb{K}^*$-surfaces. Furthermore, we show that any three-dimensi…
Given arbitrary integers $d$ and $r$ with $d \geq 4$ and $1 \leq r \leq d + 1$, a reflexive polytope $\mathcal{P} \subset \mathbb{R}^d$ of dimension $d$ with ${\rm depth} K[\mathcal{P}] = r$ for which its dual polytope $\mathcal{P}^\vee$ is normal will be constructed, where $K[\mathcal{P}]$ is the toric ring of $\mathcal{P}$.
Fano and Reflexive Polytopes from Feynman Integrals
本文研究了从准有限费曼积分中出现的Fano和反射多面体，通过计算相关双变量Ehrhart多项式确定内部点数，发现这些多面体在高维情况下稀少，并揭示了一种与Calabi-Yau流形相关的几何结构。
For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for combinatorial manifolds and, in particular, for polytopal simplicial spheres. The results are applied to a coloring problem for simplicial (or, dually, simple) polytopes which arises in the area of toric algebraic varieties.
Stanley proved that for any centrally symmetric simplicial $d$-polytope $P$ with $d\geq 3$, $g_2(P) \geq {d \choose 2}-d$. We provide a characterization of centrally symmetric $d$-polytopes with $d\geq 4$ that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.
A lattice polytope $\mathcal{P}$ is called reflexive if its dual $\mathcal{P}^\vee$ is a lattice polytope. The property that $\mathcal{P}$ is unimodularly equivalent to $\mathcal{P}^\vee$ does not hold in general, and in fact there are few examples of such polytopes. In this note, we introduce a new reflexive simplex $Q_n$ which has this property. Additionally, we show that $δ$-polynomalial of $Q_n$ is the Eulerian polynomial and show the existence of a regular, flag, unimodular triangulation.
In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n…
It is a famous open question whether every integrally closed reflexive polytope has a unimodal Ehrhart delta-vector. We generalize this question to arbitrary integrally closed lattice polytopes and we prove unimodality for the delta-vector of lattice parallelepipeds. This is the first nontrivial class of integrally closed polytopes. Moreover, we suggest a new approach to the problem for reflexive polytopes via triangulations.
New Calabi-Yau Manifolds from Genetic Algorithms
本文利用遗传算法生成自反多胞体，成功复现低维已知结果并构造出五维新自反多胞体，从而发现新的Calabi-Yau四重流形及新Hodge数。
We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.
Q-factorial Gorenstein toric Fano varieties X of dimension d with Picard number rho(X) correspond to simplicial reflexive d-polytopes with rho(X)+d vertices. Casagrande showed that any simplicial reflexive d-polytope has at most 3d vertices, if d is even, respectively, 3d-1, if d is odd. Moreover, it is known that equality for d even implies uniqueness up to unimodular equivalence. In this paper we completely classify all simplicial reflexive d-polytopes having 3d-1 vertices, corresponding to d-dimensional Q-factorial Gorenstein toric Fano varieties with Picard number 2d-1. For d even, there e…
A simplicial polytope is a polytope with all its facets being combinatorially equivalent to simplices. We deal with the edge connectivity of the graphs of simplicial polytopes. We first establish that, for any $d\ge 3$, for any $d\ge 3$, every minimum edge cut of cardinality at most $4d-7$ in such a graph is \textit{trivial}, namely it consists of all the edges incident with some vertex. A consequence of this is that, for $d\ge 3$, the graph of a simplicial $d$-polytope with minimum degree $δ$ is $\min\{δ,4d-6\}$-edge-connected. In the particular case of $d=3$, we have that every minimum edge …
A simplicial polytope is combinatorially rigid if its combinatorial structure is determined by its graded Betti numbers which are important invariant coming from combinatorial commutative algebra. We find a necessary condition to be combinatorially rigid for 3-dimensional reducible simplicial polytopes and provide some rigid reducible simplicial polytopes.
In the focus of this paper is the operation of edge contraction. One can show that simplicial 3-polytope is flag iff contraction of any its edge gives simplicial 3-polytope. Our main result states that any flag simplicial 3-polytope can be reduced to octahedron by sequence of edge contractions. Using this operation we introduce a partial order on the set of flag simplicial 3-polytopes and study Hasse graph of corresponding poset. We estimate input and output degrees of vertices of this Hasse graph.
Kalai proved that the simplicial polytopes with g_2=0 are the stacked polytopes. We characterize the g_2=1 case. Specifically, we prove that every simplicial d-polytope (d>=4) which is prime and with g_2=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d>=4) with g_2=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphe…
A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the $h$-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial $d$-polytopes, that is simplicial $d$-polytopes whose underlying graphs are $d$-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes …
Obstructions to weak decomposability for simplicial polytopes
本文证明某些单纯形多胞体甚至不是弱$O(\sqrt{d})$-可分解的，从而表明（弱）可分解性无法证明Hirsch猜想的多项式版本。
We classify the d-dimensional simplicial, terminal, and reflexive polytopes with at least 3d-2 vertices. In particular, it turns out that these are all smooth Fano polytopes. This improves on previous results of Casagrande in 2006 and Oebro in 2008. Smooth Fano polytopes play a role in algebraic geometry and mathematical physics.
We use the notions of reflexivity and of reflexive dimensions in order to introduce probability measures for lattice polytopes and initiate the investigation of their statistical properties. Examples of applications to discrete geometry include a study of randomness of self-duality of reflexive polytopes and implications for expectation values of the numbers of such polytopes in higher dimensions. We also discuss enumeration problems and related algorithms and point out interesting open problems. In this context we define the notion of IP-confined polytopes. Our new results include the list of…
We give an upper bound on the volume vol(P*) of a polytope P* dual to a d-dimensional lattice polytope P with exactly one interior lattice point, in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp, and is achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional toric Fano variety X with at worst canonical singularities.
Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.
This paper is a survey of recent advances as well as open problems in the study of face numbers of centrally symmetric simplicial polytopes and spheres. The topics discussed range from neighborliness of centrally symmetric polytopes and the upper bound theorem for centrally symmetric simplicial spheres to the generalized lower bound theorem for centrally symmetric simplicial polytopes and the lower bound conjecture for centrally symmetric simplicial spheres and manifolds.
In this paper, we consider terminal reflexive polytopes arising from finite directed graphs and study the problem of deciding which directed graphs yield smooth Fano polytopes. We show that any centrally symmetric or pseudo-symmetric smooth Fano polytopes can be obtained from directed graphs. Moreover, by using directed graphs, we provide new examples of smooth Fano polytopes whose corresponding varieties admit Kähler--Einstein metrics.
A conjecture of Kalai from 1994 posits that for an arbitrary $2\leq k\leq \lfloor d/2 \rfloor$, the combinatorial type of a simplicial $d$-polytope $P$ is uniquely determined by the $(k-1)$-skeleton of $P$ (given as an abstract simplicial complex) together with the space of affine $k$-stresses on $P$. We establish the first non-trivial case of this conjecture, namely, the case of $k=2$. We also prove that for a general $k$, Kalai's conjecture holds for the class of $k$-neighborly polytopes.
We study $n$-vertex $d$-dimensional polytopes with at most one nonsimplex facet with, say, $d+s$ vertices, called {\it almost simplicial polytopes}. We provide tight lower and upper bound theorems for these polytopes as functions of $d,n$ and $s$, thus generalizing the classical Lower Bound Theorem by Barnette and Upper Bound Theorem by McMullen, which treat the case of $s=0$. We characterize the minimizers and provide examples of maximizers, for any $d$. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of indepen…
It is proved that a certain symmetric sequence of nonnegative integers arising in the enumeration of magic squares of given size n by row sums or, equivalently, in the generating function of the Ehrhart polynomial of the polytope of doubly stochastic n by n matrices, is equal to the h-vector of a simplicial polytope and hence that it satisfies the conditions of the g-theorem. The unimodality of this sequance, which follows, was conjectured by Stanley (1983). Several generalizations are given.
Let $f_i(P)$ denote the number of $i$-dimensional faces of a convex polytope $P$. Furthermore, let $S(n,d)$ and $C(n,d)$ denote, respectively, the stacked and the cyclic $d$-dimensional polytopes on $n$ vertices. Our main result is that for every simplicial $d$-polytope $P$, if $$ f_r(S(n_1,d))\le f_r(P) \le f_r(C(n_2,d)) $$ for some integers $n_1, n_2$ and $r$, then $$ f_s(S(n_1,d))\le f_s(P) \le f_s(C(n_2,d)) $$ for all $s$ such that $r<s$. For $r=0$ these inequalities are the well-known lower and upper bound theorems for simplicial polytopes. The result is implied by a certain ``comparison …
The face lattice of any simplicial polytope is Hamiltonian
本文证明了任意单纯多胞体的面格具有哈密顿性，通过双向线壳化分解面格为立方体，并应用Gregor定理构造哈密顿圈，证实了相关猜想的主要部分。
For a d-dimensional convex lattice polytope P, a formula for the boundary volume is derived in terms of the number of boundary lattice points on the first $\floor{d/2}$ dilations of P. As an application we give a necessary and sufficient condition for a polytope to be reflexive, and derive formulae for the f-vector of a smooth polytope in dimensions 3, 4, and 5. We also give applications to reflexive order polytopes, and to the Birkhoff polytope.
We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action. As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.
In the paper we treat Gale diagrams in a combinatorial way. The interpretation allows to describe simplicial complexes which are Alexander dual to boundaries of simplicial polytopes and, more generally, to nerve-complexes of general polytopes. This technique and recent results of N.Yu.Erokhovets are combined to prove the following: Buchstaber invariant $s(P)$ of a convex polytope equals 1 if and only if $P$ is a pyramid. In general, we describe a procedure to construct polytopes with $s_R(P)>k$. The construction has purely combinatorial consequences. We also apply Gale duality to the study of …
In 1980 Provan and Billera defined the notion of weak $k$-decomposability for pure simplicial complexes. They showed the diameter of a weakly $k$-decomposable simplicial complex $Δ$ is bounded above by a polynomial function of the number of $k$-faces in $Δ$ and its dimension. For weakly 0-decomposable complexes, this bound is linear in the number of vertices and the dimension. In this paper we exhibit the first examples of non-weakly 0-decomposable simplicial polytopes.
Given a reductive group $G$ and a parabolic subgroup $P\subset G$, with maximaltorus $T$, we consider (following Dabrowski's work) the closure $X$ of a generic $T$-orbit in $G/P$, and determine in combinatorial termswhen the toric variety $X$ is $\mathbb{Q}$-Gorenstein Fano, extending in this way the classification of smooth Fano generic closures given by Voskresenski\uı and Klyachko. As an application, we apply the well known correspondence between Gorenstein Fano toric varieties and reflexive polytopes in order to exhibit which reflexive polytopes correspond to generic closures -- this list …
For a reduced word ${\bf i}$ of the longest element in the Weyl group of $\mathrm{SL}_{n+1}(\mathbb{C})$, one can associate the string cone $C_{\bf i}$ which parametrizes the dual canonical bases. In this paper, we classify all ${\bf i}$'s such that $C_{\bf i}$ is simplicial. We also prove that for any regular dominant weight $λ$ of $\mathfrak{sl}_{n+1}(\mathbb{C})$, the corresponding string polytope $Δ_{\bf i}(λ)$ is unimodularly equivalent to the Gelfand-Cetlin polytope associated to $λ$ if and only if $C_{\bf i}$ is simplicial. Thus we completely characterize Gelfand-Cetlin type string poly…
Given a (finite) simplicial complex, we define its $i$-th Laplacian polytope as the convex hull of the columns of its $i$-th Laplacian matrix. This extends Laplacian simplices of finite simple graphs, as introduced by Braun and Meyer. After studying basic properties of these polytopes, we focus on the $d$-th Laplacian polytope of the boundary of a $(d+1)$-simplex $\partial(σ_{d+1})$. If $d$ is odd, then as for graphs, the $d$-th Laplacian polytope turns out to be a $(d+1)$-simplex in this case. If $d$ is even, we show that the $d$-th Laplacian polytope of $\partial(σ_{d+1})$ is combinatorially…
Transversal numbers of simplicial polytopes, spheres, and pure complexes
本文证明了单纯复形横截数的新上下界，引入循环多胞形的“兄弟”族，并构造了横截比突破纪录的单纯球面无限族。
A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes
通过构造一个7维单纯0/1-多胞体（14个顶点）并证明其非中心对称，否定了Ziegler关于顶点数达到上界时必为中心对称交叉多胞体的猜想。
In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is reflexive if and only if $G$ is bipartite. Moreover, in the case, ${\mathcal B}_G$ has a regular unimodular triangulation. This implies that the $h^*$-polynomial of ${\mathcal B}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, namely the $γ$-positivity and the real-rootedness of the $h^*$-polynomials. In fact, if $G$ is bipartite…
For a simplicial complex K on m vertices and simplicial complexes K1,...,Km a composed simplicial complex K(K1,...,Km) is introduced. This construction generalizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of polytopes P(P1,...,Pm) defined by G. Agnarsson. The composition defines a structure of an operad on a set of finite simplicial complexes. We prove the following: (1) a composed complex K(K1,...,Km) is a simplicial sphere iff K is a simplicial sphere and Ki are the bou…
The merging operation and $(d-i)$-simplicial $i$-simple $d$-polytopes
本研究定义了一种多面体合并操作，证明其可保持$(d-i)$-单形$i$-简单$d$-多面体的性质，借此构造了多类无限多面体族，并给出了顶点数受限的该类多面体组合类型的数量下界。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。