Questions tagged [convex-polytopes]
Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
710
questions
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votes
1answer
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Tilings of lattice polytopes by transformations of lattice polytopes
A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
5
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1answer
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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 ...
3
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1answer
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Reference for “every 5-dimensional polytope has a 3-gonal or 4-gonal face”
It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.
Alternatively, I would be ...
3
votes
1answer
140 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 ...
1
vote
0answers
50 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 ...
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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,...
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votes
0answers
41 views
Separability of Minkowski Sum of well-behaved sets
Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
3
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1answer
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Is there a spherical analogue of polar duality for spherical complexes?
Let $P$ be a spherical complex, which essentially means a tiling of a sphere, let us say the $(d-1)$-dimensional sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ to fix notation, where each cell is a ...
1
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0answers
53 views
Polytopes with large dihedral angles
The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$.
The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
2
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0answers
40 views
Can a polytope with vertex-transitive edge graph or face lattice be made vertex-transitive?
Let $P\subset\Bbb R^d$ be a convex, full-dimensional polytope (convex hull of finitely many points, affine hull is the whole space), $G_P$ its edge graph and $\mathcal F_P$ its face lattice. Any of ...
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Pyramids whose volume can be computed by simple cutting and glueing
Since this question remained without answers even after a bounty, I thought it might be time to ask it here.
For which pyramid can you compute the volume from simple cut-and-glue processes? The Dehn ...
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Collection of random convex polytopes
is there either a database with h-representation of convex polytopes or is there a way to generate them? I specifically need the h-representation.
Thank you!
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0answers
26 views
An exponential integral over a closed convex polytope
For any $T\geq 2$, let us define the polyhedron $S$ given by
\begin{align*}
S:=\{\underline{t}:=(t_0,t_1,t_{2},t_{3},t_{4},t_{5},t_{6},t_{7})\in [0,+\infty)^{8}:A\underline{t}\leq (\log T)\textbf{1}\}
...
6
votes
1answer
199 views
Minimum area of the convex hull of the union of a parallelogram and a triangle
This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
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vote
0answers
33 views
fast V representation update of polytope
Say that I have both the V and the H representation of a (possibly unbounded) polytope $P$. I want to append a some rows to the H representation, how can I quickly update the V representation to ...
2
votes
1answer
59 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 $p_P(t)$ ...
2
votes
0answers
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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 $\{x\,|\,Ax\leq 0\}=\{0\}$ (is ...
1
vote
0answers
63 views
Regular triangulation of hypercube
I have started studying regular subdivisions of the $n$-cube, and came across the following post: Regularity of Delaunay triangulation of a hypercube.
My question is whether the "standard ...
1
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0answers
27 views
Definition of convex hulls via maximal sets of interior-disjoint simplices
Let the simplex cover of a finite set $\mathcal{P}\subset \mathbb{E}^n,\ n\,\le\, k:=\operatorname{card}(\mathcal{P})\,\lt\infty$ of points in Euclidean $n$-space of which no $n+1$ are co-hyperplanar,...
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votes
3answers
445 views
Number of regions formed by $n$ points in general position
Given $n$ points in $\mathbb{R}^d$ in general position, where $n\geq d+1$. For every $d$ points, form the hyperplane defined by these $d$ points. These hyperplanes cut $\mathbb{R}^d$ into several ...
3
votes
0answers
46 views
Deformations that flatten small curvature
I'm trying to show that any 3-dimensional polyhedron with many vertices can be mildly deformed so that its vertices are no longer convexly independent. I suspect it suffices to look at a vertex with ...
3
votes
0answers
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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 $n$ ...
1
vote
1answer
67 views
Interior point of a convex polytope
Suppose the convex polytope is the set of feasible solutions $\mathbf{x}\in\mathbb{R}^n$ for the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}\,,\; \mathbf{A}\in\mathbb{R}^{m\times n}$ subject to ...
2
votes
1answer
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Sampling algorithms on convex polytopes
Let $f=\mathbf{c}\cdot\mathbf{x}$ be the optimization objective function whose parameter vector $\mathbf{x}\in\mathbb{R}^n$ is subject to the following constraints in the very well-known linear-...
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votes
0answers
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How do you refer to the feasible set of solutions to a mixed-integer program?
I frequently want to refer to the feasible set of solutions to a mixed integer programming instance. Is there a name for a subset of $\mathbb{R}^n\times\{0,1\}^m$ of the form $\{(x,a)| Ax + Ba\leq b\}$...
9
votes
1answer
403 views
What is the convex hull of the quaternionic symmetries of the 3 dimensional cube?
It is well known that there are exactly five 3-dimensional regular convex polyhedra, known as the Platonic solids.
In 1852 the Swiss mathematician Ludwig Schlafli found that there are exactly six ...
16
votes
4answers
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 ...
3
votes
1answer
74 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) $\...
4
votes
2answers
366 views
addition theorems for hypersine
I learned from Wolfram MathWorld about hypersine, as being a dimensional analog trig function for hypersolid angles. There it is being defined by
The hypersine ($n$-dimensional sine function) is a ...
1
vote
1answer
61 views
Convexity of the Voronoi cells of higher-dimensional polyhedra
let a convex polytope $\mathcal{P}$ in $E^n$ be defined as in the tag-description with the additional requirement that their volume be strictly positive.
let further the Voronoi Cells $VC(f)$ of $\...
26
votes
7answers
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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 ...
2
votes
1answer
106 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 (...
4
votes
0answers
104 views
On intrinsic volumes
Let $\Gamma$ be a convex polytope in $\mathbb R^n$. The $k$-th intrinsic volume of $\Gamma$ is the number
$$
\text{v}_k(\Gamma)=\sum_{\Delta\in{\mathcal B}(\Gamma,k)}\text{vol}_k(\Delta)\psi_\Gamma(\...
3
votes
0answers
62 views
On two different descriptions of Delzant polytopes
I have seen two different ways of describing a Delzant polytope:
From Canna Da Silva https://people.math.ethz.ch/~acannas/Papers/toric.pdf, a Delzant polytope is a polytope in $\mathbb{R}^{n*}$ ...
6
votes
2answers
171 views
Cut norm versus $l_1$ norm
Let $K$ be the set of $n\times n$ matrices with zero diagonal entries and such that the sum of all entries is zero.
The cut norm of a $n\times n$ matrix $M$ is:
$$
cut(M) = \sup_{S, T, S\cap T = \...
1
vote
1answer
55 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\...
0
votes
0answers
28 views
Face computation and support function minimisation of polytope projection
Let $P \subset {\mathbb R}^m$ be a convex polytope containing 0, given by system of inequalities $Ay \leq b$. Let ${\mathbb R}^n$, $0<n<m$, be a subspace of ${\mathbb R}^m$ given by first $n$ ...
2
votes
0answers
77 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: ...
5
votes
1answer
260 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 ...
3
votes
0answers
72 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 ...
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0answers
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Vertex enumeration for polytope with a sparse halfplane description?
Say I have a (bounded convex) polytope $P\subset\mathbb R^d$ with description $Ax\le b$, where $A$ is sparse in the sense that there are at most $k$ nonzero entries in each row or column, where $k$ is ...
2
votes
0answers
54 views
A matrix that commutes with all symmetries of a vertex-transitive polytope
Let $P\subset\Bbb R^d$ be a vertex-transitive polytope aka. an orbit polytope.
Can there be a matrix $T\in\mathrm{SO}(\Bbb R^d)$ that commutes with all symmetries in $\mathrm{Aut}(P)\subset\mathrm O(\...
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0answers
47 views
How does one translate from convex hull to a set of facets (inequalities)? [duplicate]
Suppose I have defined a convex set as the convex hull of a set of points. (I know that all these points are "extremal points" of the convex set.) I know want to translate this description of the ...
0
votes
0answers
26 views
Build a (simple) polytope out of a another (simple) polytope by identification of facets
Let P be a simple $d$-dimensional simple polytope, represented by the combinatorical data of a set of facet labels $F=\{X_1, \dots, X_{|F|}\}$ and a set of vertices $V$ each being a subset of ...
4
votes
1answer
171 views
Polynomial size embeddings of toric varieties from polytopes?
Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan.
$X_P$ is always projective, because the collection of characters corresponding to the points $\...
0
votes
1answer
55 views
Overlap count of convex combination of points [closed]
Let $x_1, ..., x_n\in\mathbb{R}^d$. We know that a point $x$ in the convex hull $\text{conv}(x_1, ..., x_n)$ may be expressed as convex combinations $x=\sum_{i=1}^n r_ix_i=\sum_{i=1}^n s_ix_i$ with ...
4
votes
3answers
482 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 ...
5
votes
1answer
197 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 ...
3
votes
2answers
113 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 ...
2
votes
1answer
131 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 ...
