All Questions
Tagged with co.combinatorics convex-polytopes
244 questions
4
votes
1
answer
510
views
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 ...
- 3,472
3
votes
1
answer
258
views
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, ...
- 3,472
1
vote
0
answers
193
views
Lattice-point enumeration question involving linear combinations of matrices
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, \...
- 170
56
votes
1
answer
3k
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 $\...
- 24.7k
15
votes
3
answers
1k
views
Classification of Platonic solids
My question is very basic: where can I find a complete (and hopefully self-contained) proof of the classification of Platonic solids? In all the references that I found, they use Euler's formula $v-e+...
- 955
18
votes
1
answer
641
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 ...
- 907
1
vote
1
answer
288
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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, ...
- 3,093
4
votes
1
answer
1k
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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 ...
- 24.9k
20
votes
5
answers
1k
views
From convex polytopes to toric varieties: the constructions of Davis and Januszkiewicz
One of the most useful tools in the study of convex polytopes is to move from polytopes (through their fans) to toric varieties and see how properties of the associated toric variety reflects back on ...
- 24.7k
31
votes
4
answers
2k
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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, ...
- 37.7k
12
votes
2
answers
665
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, ...
- 85.6k
3
votes
1
answer
271
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}$ :
$$...
- 60.5k
2
votes
1
answer
303
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 ...
- 10.5k
29
votes
0
answers
3k
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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 ...
6
votes
2
answers
2k
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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 ...
- 291
7
votes
3
answers
866
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 ...
- 1,314
10
votes
3
answers
1k
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 ...
- 96.4k
22
votes
3
answers
2k
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) ...
- 28.6k
8
votes
1
answer
725
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 ...
- 902
5
votes
0
answers
503
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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 ...
- 156k
8
votes
1
answer
1k
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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!
- 81
9
votes
1
answer
1k
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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 ...
- 1,871
3
votes
0
answers
234
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 $C\in\mathcal C_d$ ...
- 17.9k
16
votes
5
answers
1k
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 $6$...
- 17.9k
11
votes
0
answers
366
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 B_{i}}{x_j}\quad\text{...
- 3,626
19
votes
2
answers
1k
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About a Delzant polytope. (In particular dodecahedron)
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 $v$ ...
- 1,037
9
votes
0
answers
738
views
Counting Lattice Points in Real Polytopes
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 ...
9
votes
1
answer
523
views
The volume of the “unit ball” in $\mathbb{R}^{m\times n}$ with respect to the cut norm
This question is inspired by the question “ε-nets with respect to the cut norm” by the user Aaron, which had been reposted to cstheory.stackexchange.com.
The cut norm ||A||C of a matrix A=(aij)∈ℝm×n ...
- 1,959
8
votes
2
answers
383
views
Do singular values of a point set determine its shape?
Suppose I have $k$ points in $d$ dimensions. Let A be a $k\times d$ matrix with $i$th row giving the coordinates of $i$th point. Do singular values of this matrix have an interpretation as some kind ...
- 2,442
3
votes
3
answers
390
views
Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
- 33
13
votes
3
answers
1k
views
Triangulations of polyhedra
A topologist came to me with this question, but everything I think should work doesn't.
How many triangulations are there of a polyhedron with n vertices?
By a "triangulation" of a polyhedron P we ...
- 1,975
5
votes
2
answers
345
views
Finding the Boundary Faces of the Zonohedron
A zonotope is a linear combination of m vectors with coefficients in [0,1]: $Z = \{ \sum \lambda_i v_i : 0 \leq \lambda _i \leq 1 \}$. The fancy way is to say it's the Minkowski sum of line segments ...
- 22.8k
6
votes
2
answers
2k
views
Birkhoff's theorem about doubly stochastic matrices
Birkhoff's theorem states:
The set of $n \times n$ doubly stochastic matrices is a convex set whose extreme points are the permutation matrices
This theorem seems to be commonly attributed to ...
- 28.6k
32
votes
0
answers
2k
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A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"
Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ .
Suppose that
(*)
For every $i \lt j \lt k$
and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$,
...
- 24.7k
5
votes
2
answers
879
views
Intersection homology for toric varieties
is there any algorithm known for computing (middle perversity)intersection homology of complex toric varieties based on their combinatorial data? I'm not looking for a computer program.
Regards,
...
- 93
19
votes
3
answers
2k
views
Cutting convex sets
Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number $...
- 17.9k
6
votes
2
answers
659
views
How many vertices of a polytope can be chopped off to produce a k-vertex facet?
Let P be a simple n-facet d-polytope with facet F, and let F have k vertices. Let H be a halfspace and Q be a simple (n-1)-facet polytope such that H ∩ Q = P.
In terms of k, what is an upper ...
- 275
12
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1
answer
651
views
Does a triangulation without fixed simplex property always exist?
Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\...
- 2,104
8
votes
2
answers
217
views
Flipping Hilbert series of semigroup rings
I'll first give intuition, and then give a precise statement.
For $|z|<1$, we have $\sum_{i \geq 0} z^i = 1/(1-z)$. For $|z|>1$, we have $\sum_{i<0} (-1) z^i=1/(1-z)$. Thus, the two ...
- 156k
0
votes
2
answers
611
views
2
votes
2
answers
2k
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Mathematical tools appropriate to analyse convex polyhedra
What mathematical tools (means: set of areas of mathematical knowledge) are appropriate to begin with to analyse (to enumerate face vectors associated with polyhedron, to calculate the combinatorial ...
22
votes
4
answers
3k
views
Can you determine whether a graph is the 1-skeleton of a polytope?
How do I test whether a given undirected graph is the 1-skeleton of a polytope?
How can I tell the dimension of a given 1-skeleton?
- 9,720
34
votes
16
answers
7k
views
Generalizations of the Birkhoff-von Neumann Theorem
The famous Birkhoff-von Neumann theorem asserts that every doubly stochastic matrix can be written as a convex combination of permutation matrices.
The question is to point out different ...
- 24.7k
13
votes
2
answers
2k
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Combinatorics of the Stasheff polytopes
First a little background for those unaware. The Stasheff polytopes (or associahedra) are certain convex polytopes that arise in the theory of $A_\infty$-algebras. There is one polytope for each $n\...
- 3,423