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 theorem'' for $f$-vectors, formulated in Section 4. Among its other consequences is a similar lower bound theorem for centrally-symmetric simplicial polytopes.
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.
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.
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 …
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…
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.
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.
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…
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.
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 …
It is known that the suspension of a simplicial complex can be realized with only one additional point. Suitable iterations of this construction generate highly symmetric simplicial complexes with various interesting combinatorial and topological properties. In particular, infinitely many non-PL spheres as well as contractible simplicial complexes with a vertex-transitive group of automorphisms can be obtained in this way.
For any flag simplicial complex $Θ$ obtained by stellar subdividing the boundary of the cross polytope in edges, we define a flag simplicial complex $Γ(Θ)$ (dependent on the sequence of subdivisions) whose $f$-vector is the $γ$-vector of $Θ$. This proves that the $γ$-vector of any such simplicial complex satisfies the Frankl-Füredi-Kalai inequalities, partially solving a conjecture by Nevo and Petersen \cite{np}. We show that when $Θ$ is the dual simplicial complex to a nestohedron, and the sequence of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that $Γ(Θ)$ is equal to…
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.
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.
A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes
Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg（马格德堡奥托·冯·古里克大学数学学院）；Institute of Mathematics, Technische Universität Berlin（柏林工业大学数学研究所）；Zuse Institute Berlin（柏林祖斯研究所）
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.
Ordinary polytopes are known as a non-simplicial generalization of the cyclic polytopes. The face vectors of ordinary polytopes are shown to be log-concave.
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 interior angle vector ($\widehatα$-vector) of a polytope is a metric analogue of the $f$-vector in which faces are weighted by their solid angle. For simplicial polytopes, Dehn-Sommerville-type relations on the $\widehatα$-vector were introduced by Sommerville (1927) and Höhn (1953). Camenga (2006) defined the $\widehatγ$-vector, a linear transformation analogous to the $h$-vector and conjectured it to be non-negative. Using tools from geometric and algebraic combinatorics, we prove this conjecture and show that the $\widehatγ$-vector increases in the first half and is flawless. In contras…
A simplicial complex $Δ$ is called flag if all minimal nonfaces of $Δ$ have at most two elements. The following are proved: First, if $Δ$ is a flag simplicial pseudomanifold of dimension $d-1$, then the graph of $Δ$ (i) is $(2d-2)$-vertex-connected and (ii) has a subgraph which is a subdivision of the graph of the $d$-dimensional cross-polytope. Second, the $h$-vector of a flag simplicial homology sphere $Δ$ of dimension $d-1$ is minimized when $Δ$ is the boundary complex of the $d$-dimensional cross-polytope.
In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three.…
The Hirsch conjecture, posed in 1957, stated that the graph of a $d$-dimensional polytope or polyhedron with $n$ facets cannot have diameter greater than $n - d$. The conjecture itself has been disproved, but what we know about the underlying question is quite scarce. Most notably, no polynomial upper bound is known for the diameters that were conjectured to be linear. In contrast, no polyhedron violating the conjecture by more than 25% is known. This paper reviews several recent attempts and progress on the question. Some work in the world of polyhedra or (more often) bounded polytopes, but s…
We consider geometric and computational measures of complexity for sets of integer vectors, asking for a qualitative difference between $f$-vectors of simplicial and general $d$-polytopes, as well as flag $f$-vectors of $d$-polytopes and regular CW $(d-1)$-spheres, for $d\ge 4$.
We investigate the question of whether any $d$-colorable simplicial $d$-polytope can be octahedralized, i.e., it can be subdivided to a $d$-dimensional geometric cross-polytopal complex. We give a positive answer in dimension $3$, with the additional property that the octahedralization introduces no new vertices on the boundary of the polytope.
In this paper we use a generating function approach to record and calculate entries of the Minkowski tensors of a polytope. We focus on ''surface tensors'', extending the methods used in arXiv:1807.10258 for moments of the uniform distribution which correspond to volume tensors. In this context we also extend the definition of the adjoint polynomial to the boundary complex of a polytope with simplicial facets. In the case of simplicial polytopes we give an explicit formulation for these surface tensors.
What is the minimal closed cone containing all $f$-vectors of cubical $d$-polytopes? We construct cubical polytopes showing that this cone, expressed in the cubical $g$-vector coordinates, contains the nonnegative $g$-orthant, thus verifying one direction of the Cubical Generalized Lower Bound Conjecture of Babson, Billera and Chan. Our polytopes also show that a natural cubical analogue of the simplicial Generalized Lower Bound Theorem does not hold.
Let P be a convex polytope not simple in general. In the focus of this paper lies a simplicial complex K_P which carries complete information about the combinatorial type of P. In the case when P is simple, K_P is the same as dP*, where P* is a polar dual polytope. Using the canonical embedding of a polytope P into nonnegative orthant, we introduce a moment-angle space Z_P for a polytope P. It is known, that in the case when P is simple the space Z_P is homeomorphic to the polyhedral product (D^2,S^1)^{K_P}. When P is not simple, we prove that the space Z_P is homotopically equivalent to the s…
De Loera et al. 2009, showed that when the rank is fixed the Ehrhart polynomial of a matroid polytope can be computed in polynomial time when the number of elements varies. A key to proving this is the fact that the number of simplicial cones in any triangulation of a tangent cone is bounded polynomially in the number of elements when the rank is fixed. The authors speculated whether or not the Ehrhart polynomial could be computed in polynomial time in terms of the number of bases, where the number of elements and rank are allowed to vary. We show here that for the uniform matroid of rank $r$ …
The class of simplicial complexes representing triangulations and subdivisions of Lawrence polytopes is closed under Alexander duality. This gives a new geometric model for oriented matroid duality.
We show that the $γ$-vector of the order complex of any polytope is the f-vector of a balanced simplicial complex. This is done by proving this statement for a subclass of Stanley's S-shellable spheres which includes all polytopes. The proof shows that certain parts of the cd-index, when specializing $c=1$ and considering the resulted polynomial in $d$, are the f-polynomials of simplicial complexes that can be colored with "few" colors. We conjecture that the cd-index of a regular CW-sphere is itself the flag f-vector of a colored simplicial complex in a certain sense.
As is well known, h-vectors of simple (or simplicial) convex polytopes are characterized. In fact, those h-vectors must satisfy Dehn-Sommerville equations and some other inequalities. Simple convex polytopes determine Gorenstein* simplicial posets and h-vectors are defined for simplicial posets. It is known that h-vectors of Gorenstein* simplicial posets must satisfy Dehn-Sommerville equations and that every component in the h-vectors must be non-negative. In this paper we will show that h-vectors of Gorenstein* simplicial posets must satisfy one more subtle condition conjectured by R. Stanley…
A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms,…
We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g_2=0 on any number of n >= 13 vertices. In particular, this implies that the ray l_1, described by Bayer (1987), is fully contained in the convex hull of all flag vectors of 4-polytopes. Especially interesting examples on 9, 10 and 11 vertices are presented.
Ordinary polytopes were introduced by Bisztriczky as a (nonsimplicial) generalization of cyclic polytopes. We show that the colex order of facets of the ordinary polytope is a shelling order. This shelling shares many nice properties with the shellings of simplicial polytopes. We also give a shallow triangulation of the ordinary polytope, and show how the shelling and the triangulation are used to compute the toric h-vector of the ordinary polytope. As one consequence, we get that the contribution from each shelling component to the h-vector is nonnegative. Another consequence is a combinatori…
A neighborliness property of marginal polytopes of hierarchical models, depending on the cardinality of the smallest non-face of the underlying simplicial complex, is shown. The case of binary variables is studied explicitly, then the general case is reduced to the binary case. A Markov basis for binary hierarchical models whose simplicial complexes is the complement of an interval is given.
In this paper, we will describe the space spanned by the angle-sums of polytopes, recorded in the alpha-vector. We will consider the angles sums of simplices and the angles sums and face numbers of simplicial polytopes and general polytopes. We will construct families of polytopes whose angle sums span the spaces of polytopes defined by the Gram and Perles equations, analogues of the Euler and Dehn-Sommerville equations. We show that the dimensions of the affine span of the space of angle sums of simplices is floor[(d-1)/2] + 1, and that of the angle sums and face numbers of simplicial polytop…
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…
Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $β$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polyto…
A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of reconstructing face lattices from partial information, first considered comprehensively in Grünbaum's 1967 book. This survey paper includes varied results and open questions by many researchers on simplicial polytopes, nearly simple polytopes, cubical polytopes, zonotopes, crosspolytopes, and Eulerian posets.
In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the upper bound~$O(n2^d)$ for the diameter of any normal pure simplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound $n-d$ if the complexes are, moreover, flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual …
The $f$-vector of a $d$-dimensional polytope $P$ stores the number of faces of each dimension. When $P$ is simplicial the Dehn--Sommerville relations condense the $f$-vector into the $g$-vector, which has length $\lceil{\frac{d+1}{2}}\rceil$. Thus, to determine the $f$-vector of $P$, we only need to know approximately half of its entries. This raises the question: Which $(\lceil{\frac{d+1}{2}}\rceil)$-subsets of the $f$-vector of a general simplicial polytope are sufficient to determine the whole $f$-vector? We prove that the answer is given by the bases of the Catalan matroid.
A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani in 1987, via a non-constructive proof using topological tools from homology theory. An elegant constructive proof was given by Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai in 2009. In this paper, we show that Kalai's conjecture holds in the particular ca…
We improve Larman's bound on the diameter of a polytope by showing that if $Δ$ is a normal simplicial complex, all of whose missing faces have size at most $r$, then the diameter of the facet-ridge graph of $Δ$ is not larger than $2^{r-2}n$, where $n$ is the number of vertices of $Δ$. We then use this result to provide new upper bounds on the diameters of the facet-ridge graphs of $k$-neighborly spheres, $r$-stacked spheres, and polytopes with small $g_r$. Specifically, our bounds imply that $r$-stacked spheres with $r=O(\log n)$ satisfy the polynomial Hirsch conjecture.
Several recent papers have addressed the problem of characterizing the $f$-vectors of cubical polytopes. This is largely motivated by the complete characterization of the $f$-vectors of simplicial polytopes given by Stanley, Billera, and Lee in 1980. Along these lines Blind and Blind have shown that unlike in the simplicial case, there are parity restrictions on the $f$-vectors of cubical polytopes. In particular, except for polygons, all even dimensional cubical polytopes must have an even number of vertices. Here this result is extended to a class of zonotopal complexes which includes simply…
