# Tagged Questions

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### Number of faces of a polytopal subdivision

Let $\mathcal{P}$ be a (bounded) polytope in $\mathbb{R}^d$ and let $\mathcal{C}$ be a polytopal subdivision of $\mathcal{P}$ [1]. Is there a known tight upper bound in the number of polytopes in ...
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### Unimodular triangulation and Ehrhart polynomials

Let $P$ be a convex lattice polytope. Then it has a polynomial Ehrhart function. I am interested in what can be said about the Ehrhart polynomial when $P$ has any of the properties is integrally ...
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### Convex hull of total orders

Let $n$ be a positive integer and $\prec$ an arbitray total order on $\{1,\dots,n\}$. I associate to this order a vector $v$ with one coordinate for every pair $(i,j)$ s.t. $1\leq i\neq j \leq n$, by ...
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### Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A ...
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### Normal polytopes - counterexample?

An integral polytope $P$ is normal if all lattice points inside the integer dilation $kP$ can be expressed as $p_1+p_2+\dots+p_k$, where $p_i \in P$ are lattice points. I am looking for an example ...
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### Classifying two-faces of four-polytopes

Motivation: This question is related to my study of hyperbolic Coxeter polytopes. In general, if one put some restrictions on the type of their dihedral angles (say, all dihedral angles are equal to ...
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### Lattice points in dilated polytopes and sumsets

Let $P$ be an integral polytope, that is, the convex hull of some points in $\mathbb{N}^d$. Let $p_1,\dots,p_m$ be all lattice points in $P$. Question: What is the condition on $P$ that guarantees ...
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### Formalization (and background) of a formula, concering the integral points of a polygon.

I have recently become aware of the following neat statement. Consider a convex polygon $P$ in the real plane with integral vertices. If we associate with every integral point $(a,b)$ the monomial ...
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### Lattice points inside a (n-dimensional) tetrahedron

Hi, overflowers. I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperplanes and ...
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### moduli space of polytopes

When considering classification problems about polytopes, I sometimes has the feeling that one need to talk about certain parametrized families, i.e. moduli space of such polytopes. But neither do I ...
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### Classification of lattice polytopes with small number of lattice points in the facets

Suppose $P$ is a convex lattice polytope in $Z^3$ without interior lattice points, and we require the interior lattice points of each facet(i.e. dimensional 2 faces) are neither too much nor too few, ...
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I would like to know some references to learn more about an answer to this question, if there are any references: Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ... 0answers 640 views ### Intersecting Family of Triangulations Let \cal T_n be the family of all triangulations on an n-gon using (n-3) non-intersecting diagonals. The number of triangulations in \cal T_n is C_{n-2} the (n-2)th Catalan number. Let ... 1answer 347 views ### Can all convex polytopes be realized with vertices on surface of convex body? The following question was asked by me on Mathematics.SE. Unfortunately, no one answered it so I thought I might give it a try one level higher. Below the line you can find the slightly edited ... 1answer 177 views ### Realizing not-quite-barycentric subdivision of a polytope Given a poset S, one can form a new poset I(S) whose elements are intervals in S (i.e. either \emptyset or [a,b] for some a\leq b\in S) with ordering by (set) inclusion. If S is ranked, ... 1answer 584 views ### Complexity of convex polytope volume calculation ? (Volume of Voronoi cell) (Error probability) Assume I have polytope in R^k given by N (k<< N) linear inequalities (A_i x < b_i). I guess complexity of its volume calculate is higher than linear in "N", am I right ? (Is the complexity ... 4answers 675 views ### The Constructions of Davis and Januszkiewicz One of the most useful tools in the study of convex polytope is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ... 4answers 1k views ### Probability of zero in a random matrix Let M(n,k) be the set of n\times n matrices of nonnegative integers such that every row and every column sums to k. Let P(n,k) be the fraction of such matrices which have no zero entries, ... 2answers 459 views ### Detecting tilings by toric geometry This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask. Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ... 1answer 181 views ### The facial structure of the convex hull of a family of characteristic functions Let S be a finite set and let \mathcal{A} \subset\mathcal{P}(S) be a family of subsets of S. Consider the convex polytope spanned by the characteristic functions of members of \mathcal{A} : ... 1answer 244 views ### Max/min problems related to associahedra or their duals (ions on balls revisited) Original motivation: This is a follow-up question to and generalization of MO Q78877 on equilibrium configurations of ions on n-Dim balls. Henry Cohn gave an excellent answer dispelling my naive ... 0answers 2k views ### Why do polytopes pop up in Lagrange inversion? I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ... 2answers 423 views ### Extreme points of transportation polytope I'm interested in n \times m joint probability tables with prescribed row and column marginals. Such tables form a convex set known as the transportation polytope. What are the extreme points of ... 3answers 625 views ### Not quite regular polyhedra Take a naive interpretation of regular polyhedra: All vertices (including epsilon ball) congruent All edges congruent All faces congruent We can now find interesting families by removing one ... 3answers 714 views ### Polytopes with few vertices. Suppose I have a convex polytope in \mathbb{R}^d which I know has few vertices (in the case which prompted this question, I seem to have a polytope in \mathbb{R}^9 which has sixteen vertices). Is ... 2answers 681 views ### Sampling from the Birkhoff polytope The set of n\times n real, nonnegative matrices whose rows and columns sum to one forms the well-known Birkhoff polytope Recently someone asked me if I knew How to sample (in polynomial time) ... 1answer 314 views ### Number of simplicial polytopes with a given f-vector Plenty of very nice literature is available on the characterization of f-vectors of simplicial complexes of diverse sorts (results by Billera, Bjoerner, Kalai, Stanley, among others). I mention, as an ... 0answers 393 views ### Reference for this polyhedral lemma Recall the definition of a fan: Let U be a finite dimensional real vector space. Then a fan is a collection \mathcal{F} of cones in U such that (1) If \sigma \in \mathcal{F} and \tau is a ... 1answer 300 views ### When is a complete fan a normal fan? Is there a characterization for when a complete fan in \mathbf{R}^n is the normal fan of a polytope? Thanks! 1answer 725 views ### Finding the vertices of a polyhedral complex coming from a GIT wall and chamber decomposition I am interested in a polyhedral/combinatorics problem that arises in algebraic geometry in the context of geometric invariant theory (GIT). Algebro-geometric background: Consider the natural ... 0answers 211 views ### Maximum of a function on d-dimensional convex compact sets Let \mathcal C_d denote the set of all d-dimensional convex compact subsets with barycenter at the origin of the d-dimensional Euclidean space \mathbb E^d. Given an element C\in\mathcal C_d ... 4answers 623 views ### Name of a polytope What is the name of the polytope \Sigma\cap (-\Sigma) for \Sigma a d-simplex with barycenter at the origin? In dimension 2, one gets a hexagon, in dimension 3 an octahedron (given by the ... 0answers 315 views ### Lower Bound on the Volume of Certain Polytopes Given a partition \rho\in\mathcal{P}(n) with k blocks$$ \rho=\{B_1,B_2,\ldots,B_{k}\} $$we can define the set of equations$$ E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in ...
Hi. I have a question. Definition. Delzant polytope $P$ is a rational convex simple polytope with the smooth condition. Here, "smooth" means that for each vertex $v$, the $n$ edges containing $v$ ...
Suppose one did have an exact formula for the number of $\mathbb{Z}^n$-lattice points intersecting an arbitrary dilate of a (not necessarily rational) finite, closed and convex $n$-polytope. As a ...