All Questions
Tagged with co.combinatorics convex-polytopes
244 questions
3
votes
1
answer
184
views
What are midway sections of simplices?
This is a (slightly modified) crosspost.
Subsequent edit - coordinates are changed to obtain simpler expressions; the existing answer is not affected.
There is a family of convex polytopes: $P_n$ is $...
- 18.4k
8
votes
2
answers
212
views
Constructing a $0/1$ polytope from an abstract simplicial complex
Let us fix $\Delta$ a finite simplicial complex, and label the vertices of $\Delta$ as $\{1,2,\ldots,n\}$. For each $F\in \Delta$ let us consider the point in $\mathbb{R}^n$ given by:
$$e_F := \sum_{i\...
- 1,889
3
votes
2
answers
438
views
If a polytope is centrally symmetric and combinatorially equivalent to a zonotope, is it a zonotope?
A zonotope is a polytope whose 2-faces are centrally symmetric.
Question: If a polytope $P$ is centrally symmetric and combinatorially equivalent to a zonotope, is it itself a zonotope?
- 13.6k
3
votes
1
answer
123
views
Maximal number of visible vertices
Let $P$ be a three-dimensional convex polytope with $N$ faces; $O$ a point outside $P$. What is the maximal number $f(N)$ of vertices of $P$ which may be seen from $O$?
- 151
0
votes
1
answer
108
views
If $C_1\subseteq C_2$ are two closed convex cones that are pointed with $\partial C_1\subseteq \partial C_2$ then is $C_1=C_2$?
Let $C_1$ and $C_2$ be two proper full dimensional closed convex cones in $\mathbb{R}^n$ that are pointed. Suppose that $C_1\subseteq C_2$ and that the boundary of $C_1$ is contained in the boundary ...
5
votes
0
answers
139
views
How does a map from permutahedra to associahedra factor through multiplihedra?
Let $P_i$ denote permutahedra, $K_i$ associahedra and $J_i$ multiplihedra. In their famous paper on operadic diagonals, Saneblidze and Umble use a projection $p_i: P_i \to K_{i+1}$ which factors as $...
- 765
10
votes
3
answers
322
views
Integer decomposition property with a partial order
Let $\mathcal{P}$ be a convex lattice polytope in $\mathbb{R}^n$. We say that $\mathcal{P}$ has the integer decomposition property (or "is IDP") if for all $k\in \mathbb{N}$ and $\alpha \in ...
- 24.2k
4
votes
3
answers
347
views
Minimal data required to determine a convex polytope
Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...
- 13.6k
2
votes
1
answer
295
views
Are orbit polytopes of rotation subgroup of Coxeter group combinatorially equivalent?
Suppose that $G\subset O(d)$ is a finite reflection (finite Coxeter) group. For any $v\in \mathbb{R}^d$ which is not fixed by any non-trivial $g\in G$, one can consider the orbit polytope (Coxeter) ...
- 439
3
votes
1
answer
159
views
Are cyclic orbitopes of permutahedra necessarily simplicies?
Suppose that $v=(v_1,\ldots, v_d)\in \mathbb{R}^d$ lies in the linear subspace $v_1+\cdots +v_d=0$, and moreover that the coordinates are pairwise distinct. The permutahedron \begin{equation} P(\...
- 439
6
votes
0
answers
381
views
An inequality related to the numbers of faces of polytopes with d+2 facets
I would like to prove an inequality related to the number of $k$-faces of two $d$-polytopes with $d+2$ facets; see (1) below.
Let $r>0$, $s>0$, $t\ge 0$, and $d\ge 2$ be such that $d=r+s+t$. We ...
5
votes
1
answer
361
views
What is known about the duals of cyclic polytopes?
What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...
- 13.6k
5
votes
2
answers
304
views
Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?
The question is in the title:
Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?
I consider only convex polytopes (convex ...
- 13.6k
1
vote
0
answers
116
views
Untruncate permutohedron of order 5
I would like to understand commutation classes of reduced expressions of the longest element in $S_5$ a little better. For this, it makes sense to look at the permutohedron of order 5. Since I am only ...
- 1,813
0
votes
0
answers
76
views
Visualization of higher Bruhat order B(5,2)
I made the following images of the higher Bruhat order B(5,2) (in the sense of Manin/Schechtman) with vZome:
image 1
image 2
image 3
Unfortunately, in vZome its not possible do have regular octagons,...
- 1,813
2
votes
1
answer
113
views
Coordinate-symmetric convex polytopes with equal Erhart (quasi)-polynomials
Recall that given a nondegenerate polytope $P \subset \mathbb{R}^n$ which is the convex set of some vectors with integral coordinates, the Erhart polynomial $p_P(t)$ a polynomial such that $p_P(t)$ ...
- 512
7
votes
1
answer
299
views
Lipschitz-continuity of convex polytopes under the Hausdorff metric
Recently, I proved the following Lipschitz-continuity like result for convex polytopes:
Let $A\in\mathbb R^{m\times n}$ and $b,b'\in\mathbb R^m$ be given such that $\{x\,:\,Ax\leq 0\}=\{0\}$ (which ...
3
votes
0
answers
70
views
On the proportion of simplicial $d$-polytopes on $n$-vertices
I have a question regarding estimates for the proportion of simplicial $d$-polytopes on $n$-vertices.
Let $c_s(n,d)$ denote the number of combinatorial types of simplicial $d$-polytopes on $n$ ...
17
votes
4
answers
1k
views
Can I build infinitely many polytopes from only finitely many prescribed facets?
Given a finite set of convex $d$-dimensional polytopes $\mathcal P$, for some $d\ge 2$.
Question: Is it true that there are only finitely many different convex $(d+1)$-dimensional polytopes whose ...
- 13.6k
3
votes
1
answer
93
views
Affine equivalence of Coxeter permutahedra?
Suppose that $W=\langle s_1,\ldots, s_d\mid (s_is_j)^{m_{ij}}=e\rangle$ is a finite reflection group and consider its standard $d$-dimensional geometric realization (i.e., the Tits representation) $\...
- 157
31
votes
7
answers
3k
views
Why are we interested in permutahedra, associahedra, cyclohedra, ...?
The following families of polytopes have received a lot of attention:
permutahedra,
associahedra,
cyclohedra,
...
My question is simple: Why?
As I understand, at least the latter two were ...
- 13.6k
2
votes
1
answer
153
views
Number of orthants intersected by a convex hull
I'm trying to figure out the following problem:
Let $x_1,\ldots,x_k\in\mathbb{R}^n$ be some points for some $k<n$. Let $\mbox{conv} (x_1,\ldots,x_k)$ be their convex hull. I'm looking for a tight (...
- 23
7
votes
2
answers
552
views
Cut norm versus $l_1$ norm
Let $K$ be the set of $n\times n$ matrices with zero diagonal entries and such that the sum of all entries is zero.
The cut norm of a $n\times n$ matrix $M$ is:
$$
cut(M) = \sup_{S, T, S\cap T = \...
- 2,772
1
vote
1
answer
112
views
Regions of hyperplane arrangements and their faces
Consider a finite hyperplane arrangement $\mathcal{A}$ over $\mathbb{R}^n$. Let the regions given by $\mathcal{A}$ be $\mathcal{R}(\mathcal{A})=\{A_1,\dots A_m\}$ for some $m$.
For any index set $I\...
- 157
2
votes
0
answers
94
views
Anything similar to cone product formula (for convex polytopes)?
The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula
$$
C(U) C(V) = C(J(U, V)) + DUV
$$
where
$$
J(U, V) = U C(V) + C(U) V - e_1 UV
$$
is the join formula.
Note: ...
- 459
6
votes
1
answer
539
views
Proofs of Euler's characteristic formula for n-Dim polytopes
Twenty proofs of Euler's formula V - E + F - 1 = 1, which applies to convex polyhedrons, i.e., 3-dimensional polytopes, are presented at the Geometry Junkyard.
I'm interested in proofs of the more ...
- 10.5k
3
votes
0
answers
86
views
Reference on the faces of the circulation polytope
On page 4 of Generating all vertices of a polyhedron is hard it is mentioned that the facial structure of the circulation polyhedron* is well understood. I am trying to find a reference for this.
I ...
- 1,462
4
votes
3
answers
617
views
An example of a "simple poset" which does not belong to a convex polytope
Are there examples of d-regular graphs (i.e. graphs where every node has exactly d adjacent nodes) which are not the 1-skeleton of a simple convex polytope?
UPDATE:
New version of the question: is ...
- 1,435
5
votes
1
answer
280
views
Classification of vertex-transitive zonotopes
Zonotopes are convex polytopes that can be defined in several equivalent ways:
parallel projections of cubes,
Minkowsi sums of line segments,
only centrally symmetric faces,
...
I wonder whether ...
- 13.6k
4
votes
2
answers
173
views
4-polytopes with only one kind of regular facet
Is there a neat way to show (or a reference that already proves) that
the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
the 24-cell is the only convex 4-polytope in ...
- 13.6k
3
votes
1
answer
152
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
- 13.6k
5
votes
1
answer
417
views
Iterated derivative and rectangular standard Young tableaux
We first make a few definitions, seemingly out of the blue (they are introduced/defined in this paper).
Let $F^0_{a}(z) = (1-z)^{-1}$ and define recursively
$$
F^{k+1}_{a}(z) = z^{a-1} \frac{d^a}{dz^...
- 15.8k
3
votes
0
answers
264
views
Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)
The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...
- 10.5k
8
votes
3
answers
390
views
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...
- 13.6k
1
vote
0
answers
49
views
Realizing 0/1-polytopes with shortest possible edge lengths
Has there been something written about the following question?
Question: Given a 0/1-polytope, what is the shortest edge lengths with which this polytope can be realized as a 0/1-polytope.
The ...
- 13.6k
1
vote
1
answer
103
views
A source for $01$-polytopes
Can you recommend any books or survey articles on $01$-polytopes, thats is, polytopes with vertices in $\{0,1\}^n$?
I am less interested in random $01$-polytopes, but more in the combinatorial ...
- 13.6k
2
votes
1
answer
85
views
The number of Hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$
Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct ...
- 3,871
3
votes
1
answer
94
views
Regular triangulations of star-convex polyhedra with given boundary
Given an $n$-dimensional star-convex polyhedron $P\subset \mathbb{R}^n$ with simplicial facets, is it always possible to construct a regular triangulation $K$ of $P$ which does not subdivide the ...
- 165
1
vote
1
answer
111
views
Given $H_{N}=\{\vec{x} \in [-1,1]^N:\sum_{i=1}^N x_i = 0\}$, what is the smallest subset $S \subset H_{2N}$ such that $\mathrm{conv}(S)=H_{2N}$
Motivation:
This is related to a different question I asked in April. It occurred to me while thinking about the sums of uniform random variables and it stuck in my mind because it's the special case ...
- 3,871
5
votes
1
answer
322
views
Sufficient criterion for a simplicial sphere to be polytopal
Are there any purely combinatorial criteria which allow you to deduce that a spherical simplicial complex is polytopal (i.e., there exists a simplicial polytope whose boundary is isomorphic to it)?
...
- 6,292
11
votes
1
answer
228
views
Geometric realization of combinatorial self-duality in polytopes
Let's say I have a combinatorially self-dual polytope $P\subseteq\Bbb R^d$, i.e., its face lattice is isomorphic to its dual (you reverse the direction of the lattice order).
Question: Is it always ...
- 13.6k
2
votes
0
answers
252
views
Understanding the geometry of $H_{n}=\{\vec{x} \in [-N,N]^n:\sum_{i=1}^n x_i = 0\}$
I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $:
...
- 3,871
1
vote
1
answer
125
views
Projections of particular simplex yielding boundary of a regular polygon?
What is the maximum $m$ such that there is a simplex with $n$ vertex points in $n-1$ dimensions whose projection yields boundary of a regular $m$-gon on $2D$ plane?
- 1,826
5
votes
0
answers
140
views
$q$-Kostant partition function and flow polytopes?
The Kostant partition function is known to be related to volumes and Ehrhart polynomials of flow polytopes of graphs (see e.g. https://link.springer.com/article/10.1007/s00031-008-9019-8 or https://...
- 24.2k
1
vote
0
answers
79
views
What is known about the combinatorics of the hyperplane arrangement spanned by cyclic polytopes?
Let $1\leq d$ be an integer.
Consider the $d$-dimensional moment curve $\mu\colon \mathbb R\to \mathbb R^d$ given by $t\mapsto (t,t^2,\dots, t^d)$. Given a finite subset $S\subset \mathbb R$ of ...
- 725
12
votes
2
answers
510
views
Positivity of the coefficients of the Ehrhart polynomial of a cross-polytope
Question 35996 asks about the Ehrhart polynomial $i_d(n)$ of the
standard regular cross-polytope. It can be defined equivalently by
$$ \sum_{n\geq 0}i_d(n)x^n = \frac{(1+x)^d}{(1-x)^{d+1}}. $$
It ...
- 50.8k
3
votes
2
answers
344
views
Is a vertex- and edge-transitive polytope already a uniform polytope?
I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive).
Question: Is every such ...
- 13.6k
2
votes
0
answers
87
views
Existence of a "generic enough" lattice point interior to a lattice triangle
Let $T$ be a lattice triangle in $\Bbb R^2$ (i.e. the convex hull of three noncolinear points in $\Bbb Z^2$), and assume it has at least one interior lattice point. Is it always possible to find a ...
- 3,079
5
votes
1
answer
349
views
What is the lower bound for the number of facets that a general convex $d$-polytope with $n$ vertices can have?
I am familiar with Barnette's Lower Bound Theorem on the number of facets a $d$-dimensional simplicial convex polytope with $n$ vertices can have. Is there a similar result for a general (i.e. not ...
- 53
0
votes
0
answers
42
views
When is the set of faces of a convex polytope algebraically independent?
This is related to another question of mine
Let $V=\Bbb R^n$. Morelli defined the (commutative unital) ring $L(V)$ to be the additive group generated by the indicator functions of convex polytopes in ...
- 3,079