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
942 questions
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Are Minkowski sums of upward closed "convex" sets in $\mathbb{N}^k$ still "convex"? (WAS: Comparing mana costs in Magic: The Gathering)
This was originally a question about comparing mana costs in Magic: The Gathering, but it's turned into a question about Minkowski sums of upward-closed convex sets in $\mathbb{N}^k$. The original ...
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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) $\...
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On Kalai's $3^{d}$ conjecture
I just learned the existence of Gil Kalai's $3^{d}$ conjecture, which according to Wikipedia, is proven for $d$ at most $4$. It states that every $d$ dimensional polytope with central symmetry has at ...
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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 ...
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On intrinsic volumes
Let $\Gamma$ be a convex polytope in $\mathbb R^n$. The $k$-th intrinsic volume of $\Gamma$ is the number
$$
\text{v}_k(\Gamma)=\sum_{\Delta\in{\mathcal B}(\Gamma,k)}\text{vol}_k(\Delta)\psi_\Gamma(\...
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Monge-Ampère measures and Kazarnovskii pseudovolume
Let $\Gamma\subset\mathbb C^n$ be a convex polytope and let $h_\Gamma(z)=\max_{v\in\Gamma}{\rm Re}\langle z,v\rangle$ be its support function with respect to the standard scalar product on $\mathbb C^...
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Mathematical Structure and Objects Induced by Pairs of Disjoint Subsets
Let $\mathcal{S}$ be a finite, discrete and non-empty set, i.e.,
$$\begin{align} \operatorname{card}\left(\mathcal{S}\right) & =:n\in\mathbb{N}^+\\ V& :=\{v\subset\mathcal{S}\ |\ v\ne\...
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How different can the constituents of an Ehrhart quasi-polynomial be?
Consider a $d$-dimensional convex rational polytope $P\subset\mathbb{Q}^d\subset\mathbb{R}^d$. Then, it's a standard fact that in general the function counting the number of lattice points inside the ...
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Sampling uniformly from the vertices of a polytope
I'm looking for a reference on how to sample uniformly (and preferably efficiently, elegantly, etc.) from the vertices of a polytope. I gather that enumerating vertices is hard. I also note the MO ...
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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 ...
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Can we combine the symmetries of two polytopes to create a more symmetric polytope?
Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$.
The symmetry group $\mathrm{Aut}(P_i)\subset\...
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How to find the vertices of the set $\{v_i\in \mathbb{R}:a_1\ge v_1\ge v_2\ge \cdots\ge v_n\ge 0,\ q_2\le \sum_{i=1}^n p_iv_i\le q_1\}$
I am given a set of inequalities $v_1\ge v_2\ge \cdots\ge v_n\ge 0$, $q_2\le \sum_{i=1}^n p_iv_i\le q_1$, with $\{p_i\}_{i=1}^n,\ q_1,q_2$ positive reals, and only one bound for the coordinates: $v_1\...
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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)?
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Given the vertices of a convex polytope, how can we construct its half-space representation?
Let us say I have the vertices of a polytope $V = \{v_1,\dots,v_k\} \subset \mathbb R^n$. Is it possible to write $V$ as intersection of half-spaces using the information from the vertices, i.e., can ...
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Integral representations of finite groups and lattice point geometry
See the edit at the bottom (April 2021)
This contains both a reference request, and a specific problem.
Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group ...
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Fast projection onto a subspace
Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
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Pictures of the von Neumann polytope
Are there any graphic portrayals of von Neumann polytopes in low dimensions?
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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 ...
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Ellipse naturally associated with a polygon
My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...
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How many unit simplices are needed to cover a unit $n$-cube?
The volume of an $n$-dimensional simplex of unit edge length is
$$V(n) = \frac{\sqrt{n+1}}{n! 2^{n/2}} \;,$$
so at least $\lceil 1/V(n) \rceil$ such simplices are needed to cover the unit $n$-cube.
...
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Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?
In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...
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Area of cross-section (at midpoint perpendicular to longest diagonal) in the unit cube of dimension N
Take a unit cube (of side 1) in N dimensions. Construct the cross-section at the midpoint of the longest diagonal. What is the area of this N-1 dimensional region? I can compute this, but it would be ...
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Which (semi)regular polyhedra are combinations of two others?
The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and $...
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4-polytope with vertices at the binary octahedral group
Does anybody know if there is a convex polytope in $R^4$ with vertices at the binary octahedral group (identifying $H$ with $R^4$).
The binary tetrahedral group lies at the vertices of the so-called ...
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Parallelepiped is defined by the volumes of its faces
Let $v_1,...,v_n\in \mathbb{R}^n$ be linearly independent. The parallelepiped defined by these vectors is $P(v_1,...,v_n)=\{\sum_{i=1}^{n}\alpha_i v_i|~0\le\alpha_i\le 1\}$. Observe that while the ...
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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 ...
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Does quantifier elimination help here?
Suppose we have a quantified linear program
$$\exists z_1,\dots,z_{poly(n)}\in\mathbb R$$
$$\exists u_1,\dots,u_n\in\mathcal P\cap\mathbb R^m$$
$$\forall v_1,\dots,v_n\in\mathcal P\cap\mathbb R^m$$
$$...
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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-...
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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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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 ...
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Is the preimage of a face under an affine map a face?
Let $X\subseteq\mathbb{R}^n$ be a convex set. Let $\pi{:}\ X\to\mathbb{R}^m$ be a linear map, with $m<n$ (for example, a projection). Let $\pi^{-1}(y)=\{x\in X\mid\pi(x)=y\}$ denote the inverse of $...
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Convex Sets and Nearest Neighbors
For a set $S \subseteq \mathbb{R}^n$ and a point $x \in \mathbb{R}^n$, let $c_S(x)$ be the point $s \in S$ that minimizes $\|x-s\|$ if such a point exists and is unique. It is known that $c(x) = s$ ...
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What importance does the Hirsch conjecture have to Simplex Complexity?
The Hirsch conjecture asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional convex polytope with $n$ facets has diameter at most $n - d$.
After being open for decades, Francisco Santos has ...
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Measure-minimizing simplex with fixed inradius
Let $\Delta^n$ be an $n$-simplex in $\mathbb{R}^n$. Let $V$ be the volume and $r$ the inradius (radius of the inscribed sphere) of $\Delta^n$. There is a well-known result that
$$
V \geq \frac{n^{n/2}(...
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non-convex Polytope definition
I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex polytope in the following way:
$$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, \...
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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 ...
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Complexity of scissors congruence?
Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
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The stock market polytope: explanation?
Ovidiu Racorean.
"Crossing stocks and the positive Grassmannian I: The Geometry behind Stock
Market."
(arXiv Abstract link)
Anyone care to offer a summary of what's going on here?
(The ...
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Repeating an operation infinitely makes any convex $n$-gon a regular $n$-gon?
For any convex $n$-gon $P_{0,1}P_{0,2}\cdots P_{0,n}$, let us consider the following operation :
Operation : Let $k=0,1,\cdots$. Take $n$ points $P_{k+1,i}\ (i=1,2,\cdots,n)$ outside of $n$-gon $P_{...
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The $32$-deg polynomial for the tetrahedron inscribed in the icosahedron?
This MO answer discusses this table involving the maximal side lengths of the five Platonic solids $T,C,O,D,I$ inscribed in the other solids,
This table is also found in Moritz Firsching's paper. I ...
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A question about polytopes related to linear programming
Given linear functions $f_1({\bf x}),\dots,f_n({\bf x})$ on ${\bf R}^m$, let $K = \{(a_1,\dots,a_n) \in {\bf R}^n:$ the $n$ halfspaces $\{{\bf x}: f_i({\bf x}) \leq a_i\} $ have nonempty intersection$\...
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The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool
In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
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Approximation of convex hull in high dimension
What are efficient methods (polytime) to compute an approximation of the convex hull in high dimension (say, $30000$) for a given set of points?
Edit:
I am looking for an algorithm for getting the ...
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On maximal regular polyhedra inscribed in a regular polyhedron
Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1.
For example, it's easy to prove that ...
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Information needed to distinguish combinatorially isomorphic polytopes (up to affine equivalence)
I originally posted this question on Stack Exchange, thinking it perhaps does not qualify as "research-level" but it received no answers... hopefully someone here can help.
The title pretty much ...
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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^\...
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Break polyhedron into tetrahedron
Given a polyhedron consists of a list of vertices (v), a list of edges (e), and a list of surfaces connecting those edges (...
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Example of convex n-gon that cannot be decomposed into k congruent convex polygons
I asked a related question here on MO without any answers yet.
The question is in the title - give an example of a convex $n$-gon that cannot be subdivided into $k>1$ congruent convex polygons.
...
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The set of polytopes with given $f$-vector
Let $f=(f_0,\ldots f_n)$ be a vector in $\Bbb N^{n+1}$. Let $X$ be the set of all (ordered) $f_0$-tuples in $\Bbb R^n$ whose convex hull has $f$ as its $f$-vector. Assume that $X$ is non-empty. Is ...
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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 $p_P(t)$ ...
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