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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

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Selecting vertices in a convex polygon

Given $n$ vertices of a convex polygon in $\mathbb{R}^2$, selecting two points that are furthest apart is done by finding the diameter in a convex polygon. But how can one select three vertices such ...
maths123456's user avatar
8 votes
1 answer
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The polytope algebras generated by polytopes with rational vs arbitrary vertices

The polytope algebra was defined by P. McMullen in "The polytope algebra" Adv. Math. 78 (1989) as follows. Let us denote by $\Pi'_\mathbb{R}$ the quotient of the free abelian group generated ...
asv's user avatar
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testing whether a polyhedral complex is convex

Definitions A (polyhedral) cone in $\Bbb R^n$ is the solution set of a finite number of inequalities of the form $a_1x_1+\cdots+a_nx_n\geq 0$. Note that I don't require strict convexity, i.e. a cone $...
Avi Steiner's user avatar
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1 vote
0 answers
71 views

Diminishing of the $4_{21}$

One of the projections of the $4_{21}$ polytope (https://en.m.wikipedia.org/wiki/4_21_polytope) into four dimensions positions its vertices as those of two concentric 600-cells scaled by the golden ...
Daniel Sebald's user avatar
1 vote
1 answer
139 views

The relationship between facets of an inscribed polytope and those facets' shadows

I posted this question thinking that the response would be two or three answers that say "Counterexamples to this are found in every textbook—for example this one and this one and this one." ...
Michael Hardy's user avatar
1 vote
1 answer
337 views

High-dimensional polytopes

I have two questions regarding polytopes in high dimensions, $\mathbb{R}^d$ where $d > 3$, that I could not find resources for on the web for. Suppose I have a polytope that is non-convex: How can ...
GuyS's user avatar
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3 votes
1 answer
171 views

Covering radius of a lattice from relevant vectors

Let $L$ be an $n-$dimensional lattice. The Voronoi region of $L$ is given by $$ \mathcal{V}(L)=\big\{x\in\mathbb{R}^n~|~ \|x\|_2\leq \|x-v\|_2~\forall v\in L\setminus\{0\}\big\}. $$ Considering the ...
FermaX's user avatar
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3 votes
1 answer
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What property of ranked poset ensures that it is determined by its vertex-facet incidences?

For a convex polytope, its face poset is combinatorially determined by vertex-facet incidences. Now suppose we have an arbitrary finite poset that is ranked, so I can still speak of vertices and ...
Dasha Poliakova's user avatar
4 votes
1 answer
274 views

"Permutation matrix" but non-zero entries are replaced by $e^{ix}$

Birkhoff–von Neumann theorem states that a polytope formed by a set of doubly-stochastic matrices has extreme points that are permutation matrices. I am wondering if there is a similar theorem for a ...
CWC's user avatar
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3 votes
0 answers
130 views

Counting lattice polytopes by volume

For any $n \in \mathbb{N}$ and $B \in \mathbb{R}_{\geq 0}$, let $\mathcal{P}(n,B)$ be the set of $n$-dimensional convex polytopes $\Delta \subseteq \mathbb{R}^n$, taken up to integral, unimodular ...
Riccardo Pengo's user avatar
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0 answers
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Generalisation of spanning tree in simplex

This is a question I asked on Stackexchange (https://math.stackexchange.com/q/4004734/400240) but I did not receive an answer (I think it was too hard). Per someone's suggestion I put it here. Let $\...
IAnemaet's user avatar
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1 vote
0 answers
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refining a coherent triangulation

I am relatively new to this topic, so this question may be easy/naive to some experts. Here goes.. I have a finite set of points $S\subset\mathbb R^2$ (you may increase the dimension of the ambient ...
Jose Capco's user avatar
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2 votes
1 answer
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Definition of "regular" in Stringham's "Regular figures in n-dimensional space"

I've been reading Irving Stringham's 1880 thesis, "Regular Figures in n-dimensional Space" (only 14 pages!), after it was mentioned by Coxeter in Regular Polytopes (§7.x). I'm confused about ...
Nick Matteo's user avatar
24 votes
1 answer
1k views

Which unfoldings of the $d$-dimensional hypercube tile $(d{-}1)$-space?

A six year old question, Which unfoldings of the hypercube tile $3$-space?, has just been answered by Moritz Firsching: All $261$ unfoldings tile space! So now we know: For $d=2$, the unfolding of ...
Joseph O'Rourke's user avatar
3 votes
0 answers
111 views

"Slim" directed polytopes: any established name for them?

This is a "looking for context" question. Let's say that a polytope is directed if its 1-skeleton is an oriented graph with no cycles, one source, one sink. (Edit: let us additionally assume ...
Dasha Poliakova's user avatar
1 vote
0 answers
63 views

Counting $\bmod 2$ number of vertices of sparsely represented polyhedra

Given a polyhedron $$Ax\geq b$$ is there an $NC^1$ or an $NC^2$ algorithm to count the number of vertices $\bmod2$? Assume $A\in\{0,1\}^{m\times n}$ and $b\in\mathbb Z^{m}$ ($m=O(n)$) and assume rows ...
Turbo's user avatar
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3 votes
0 answers
50 views

Modular counting of integral points under sparse non-negativity

Given a polyhedron $$Ax\geq b$$ where every entry of $A,b$ are non-negative and $A\in\{0,1\}^{m\times n}$ and there are $O(1)$ (say $\leq8$) non-negative entries per row of $A$ is it possible to ...
Turbo's user avatar
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1 vote
0 answers
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Detecting non-negativity of a single constraint by polyhedral constraints - $II$

Let $$\langle a,x\rangle=b$$ be a linear constraint where $x\in\mathbb R^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\mathbb Z_{\...
Turbo's user avatar
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0 votes
1 answer
110 views

Detecting non-negativity of a single constraint by polyhedral constraints - $I$

We consider $$\langle a,x\rangle=b$$ (linear constraints) where $x\in\mathbb R_{\geq0}^n$ and every entry in $a=(a_1,\dots,a_n)$ is in $\mathbb Z_{\geq0}^{n}$ (non-negative) and the entry $b$ is in $\...
Turbo's user avatar
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2 votes
0 answers
54 views

Inverting "codimension matrix" for polytopes?

Let $P$ be an abstract polytope. Let's construct its square matrix $A$ as follows. Its lines and columns are labelled by all faces of $P$, of all dimensions. Put $A(F_1,F_2)=t^m$ if $F_1$ is a subface ...
Dasha Poliakova's user avatar
3 votes
0 answers
154 views

Ehrhart-Macdonald reciprocity with multiplicities

Let $P$ be a convex lattice polytope in $\mathbb{R}^n$. The function $L(t, P) = |\mathbb{Z}^n \cap t\cdot P|$ is a polynomial, and we have an equality $$L(-t, P) = (-1)^nL(t, P^{int}),$$ where $P^{int}...
Rybin Dmitry's user avatar
0 votes
0 answers
98 views

Asymptotic optimal sphericity

How quickly does maximum sphericity of polyhedra with $n$ faces approach 1 as $n→∞$? I can show that sphericity $1 - \frac{5 \sqrt{3} π}{27n} - O(n^{-3/2})$ is possible. Is this, especially $O(n^{-3/...
Dmytro Taranovsky's user avatar
1 vote
0 answers
122 views

Integrality of polyhedra

Given two polyhedra in $H$ representation $P_1:Ax\leq b$ and $P_2:Bx\leq c$ which are integral are bounded when is their intersection also integral? Given two polyhedra in $H$ representation $P_1:Ax\...
Turbo's user avatar
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2 votes
0 answers
106 views

How to judge whether the following convex set contains a given point?

Let the set $\mathcal{S}=\left\{ \sum_{i=1}^n x_i\mathbf{h}_i:x_i\in[0,1] \text{ for all }i\right\}\subset\mathbb{R}^r$, i.e., a zonotope generated by $n$ column vector $\mathbf{h_1},\cdots,\mathbf{h}...
RyanChan's user avatar
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2 votes
0 answers
93 views

An easy way to recognize the edges of an orbit polytope?

Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is $$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...
M. Winter's user avatar
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0 votes
0 answers
93 views

Number of vertices in a polyhedron

Consider polytopes $$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$ $$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$ $$B[z_{1},z_{2},z]'\leq c$$ having vertex count $v_1,v_2$ and $v$ respectively. We ...
Turbo's user avatar
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4 votes
1 answer
290 views

Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities: $$ \left<x,u_F\right> \geq -a_F$$ where $u_F\in \...
Luis Ferroni's user avatar
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1 vote
0 answers
36 views

How to express a polytope by a matrix inequalty? [duplicate]

Consider a convex V-polytope generated by the origin and $n$ points $\mathbf{h}_1,\cdots,\mathbf{h}_n$ in $\mathbb{R}^r$. A Theorem in the area of convex geometry shows that each V-polytope is a H-...
RyanChan's user avatar
  • 550
2 votes
0 answers
138 views

Is the projective symmetry group of a polytope more general than its linear symmetry group?

Give a (convex) polytope $P\subset\Bbb R^d$ (the convex hull of finitely many points). Consider its linear and projective symmetry groups: \begin{align} \DeclareMathOperator{\Aut}{Aut} \...
M. Winter's user avatar
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3 votes
0 answers
58 views

Classifying/enumerating vertex-transitive simplicial polytopes

I'm interested in understanding the class of simplicial polytopes in $\mathbb R^n$ whose Euclidean isometry group $G$ acts transitively on the vertices. These are examples that I know of: simplicial ...
Brent Kerby's user avatar
4 votes
1 answer
149 views

A combinatorial characterization of the central inversion of a polytope?

Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...
M. Winter's user avatar
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0 votes
4 answers
457 views

Confining a polytope to one side of an affine hyperplane

Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem. This answer on math.stackexchange.com claims the ...
Hans's user avatar
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4 votes
0 answers
130 views

"Baues poset" of shellings of simplicial polytope?

Let me start with some background I want to use as analogy. Consider a (convex) polytope $P$ and its set of triangulations. Among all the triangulations, a well behaved subset are the regular ones: ...
Sam Hopkins's user avatar
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16 votes
1 answer
508 views

Can solutions to Thomson's problem have pentagons?

Thomson's problem asks for the minimum-energy configuration for $N$ electrons on a sphere (refs: https://en.wikipedia.org/wiki/Thomson_problem, https://sites.google.com/site/adilmmughal/...
Alex Meiburg's user avatar
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1 vote
0 answers
323 views

Decomposition of Polyhedral - An example

There is no doubt that clear examples consolidate the understanding of concepts being learnt. I am new to finding the structure and decomposition of a polyhedra. Suppose that we have the system $$ \...
holala's user avatar
  • 111
1 vote
1 answer
1k views

Check if a point is in the interior of the convex hull of some other points in high dimensions, and lower-bounding the largest enclosed ball [closed]

Given $m$ points $P=\{p_0, p_1, ..., p_m\}$ in high dimensions (e.g. 100), it is known that computing (or even representing) their convex hull $\text{conv}(P)$ is generally intractable due to the ...
Dazheng's user avatar
  • 11
6 votes
2 answers
185 views

Partitions of convex planar regions into zonogons

A zonogon is a centrally symmetric convex polygon. Are there convex non-zonogons that can be partitioned into a finite number of (convex) zonogons? Same as 1 with the pieces allowed to be nonconvex ...
Nandakumar R's user avatar
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7 votes
1 answer
313 views

Iterating projections to random halfspaces

Consider the following process: Start with a set $S = \mathbb R^n$. Repeat $L$ times: choose a random orthonormal basis $u_1, \ldots, u_n$, and consider the cone $C = \{ \sum \alpha_i u_i : \alpha_i \...
Daniel Paleka's user avatar
3 votes
0 answers
87 views

Additional symmetries of the Traveling Salesman Polytope

Given the complete graph $K_n=(V,E)$, the Traveling Salesman Polytope is a convex polytope in $\Bbb R^E$ obtained as the convex hull of the indicator vectors of (edge-sets of) Hamiltonian cycles in $...
M. Winter's user avatar
  • 13.6k
2 votes
1 answer
119 views

Anchor sets for lattice polygons: Part I

Suppose $V=\{(x_1,y_1), (x_2,y_2),\dots,(x_v,y_v)\}$ is a vertex set of lattice points satisfying $$0=x_1<x_2<\dots<x_v \qquad \text{and} \qquad y_1>y_2>\cdots>y_{v-1}>y_v=0.$$ ...
T. Amdeberhan's user avatar
2 votes
1 answer
106 views

Are zonotopes determined by their edge-graph?

General polytopes are not determined by their edge-graph (up to combinatorial equivalence). But I came accross the statement that zonotopes are determined in this way. Question: Is this true? And ...
M. Winter's user avatar
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3 votes
1 answer
336 views

Cohomology ring of a hypersurface in toric variety

Let $X^n$ be a smooth projective toric variety corresponding to a simple polytope $P$. It is well known that the cohomology ring $H^*(X)$ can be described in terms of combinatorics of faces of $P$. ...
asv's user avatar
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0 votes
1 answer
97 views

If there are eigenvectors with largest components $i$ resp. $j$, then is there an eigenvector with two largest components $i$ and $j$?

Let $G=(V,E)$ be a connected (finite simple) graph with vertex set $V=\{1,...,n\}$ and let $\theta_2\in\Bbb R$ be the second-largest eigenvalue of its adjacency matrix. I wonder about the following ...
M. Winter's user avatar
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15 votes
2 answers
481 views

Ehrhart period collapse for $123\ldots k$-avoiding Birkhoff polytope?

For $1 \leq r \leq n$, let $\mathcal{B}^n_r$ denote the polytope of all real matrices $$ \pi = \begin{pmatrix} \pi_{1,1} & \pi_{1,2} & \cdots & \pi_{1,n} \\ \pi_{2,1} & \ddots & \...
Sam Hopkins's user avatar
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4 votes
2 answers
190 views

Existence of a fundamental domain for the convex hull of group action on a rational polytope

Let $P \subset \mathbb R^n$ be a compact rational polytope (the affine space spanned by $P$ may not be $\mathbb R^n$). Let $G \subset {\bf GL}(n,\mathbb Z)$ be an arithmetic group. Let $$C = {\rm Conv}...
Li Yutong's user avatar
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2 votes
1 answer
128 views

State-of-the-art article on "uniform 5-polytopes?"

I would like to read article(s) that provide the “state of the art” on the following open problem: “Enumerate all convex uniform 5-polytopes.” This problem is posted on the “Open Problem Garden” (http:...
math4's user avatar
  • 155
9 votes
2 answers
341 views

Are there centrally-symmetric self-dual polytopes in dimension $d> 4$?

A convex polytope $P\subset\Bbb R^d$ is centrally symmetric if $-P=P$. It is self-dual (or better, self-polar?) if its polar dual $P^\circ$ is congruent to $P$, that is, there is a map $X\in\mathrm O(\...
M. Winter's user avatar
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0 votes
0 answers
54 views

Attached convex "hulls"

Let $\mathcal{P}$ a finite set of points of a Euclidean $\mathbb{E}^n$ and take the union $\mathrm{U}(\mathcal{P})$ of all closed half-spaces defined by $n$ elements of $\mathcal{P}$ that contain only ...
Manfred Weis's user avatar
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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 $...
მამუკა ჯიბლაძე's user avatar
8 votes
2 answers
213 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\...
Luis Ferroni's user avatar
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