# 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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### How to characterize the action of the central inversion on the face lattice of a polytope?

Given a centrally symmetric polytope $P\subset\Bbb R^d$.
The central inversion $-I$ induces an automorphism $\phi$ on the face lattice $\mathcal F(P)$ of $P$.
Question: Is there a way to see that ...

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### 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 to be a complex problem.
This answer on math.stackexchange.com claims the ...

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### “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: ...

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### On number of facets in projected polytopes of certain sparseness

$\DeclareMathOperator\Set{Set}$
Let
$Set=\{x\in\mathbb Z^{n}:\exists y\in\mathbb Z^m\text{ satisfying } A[x,y]'\leq b\}$
where $A$ has $r=km$ rows and $k=O(1)$.
I am trying to write
$$
Set=\{x\in\...

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### 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/...

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### 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
$$ \...

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### 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 ...

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### 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 ...

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### 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 \...

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### 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 $...

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### 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
$$0=x_1<x_2<\dots<x_v \qquad \text{and} \qquad y_1>y_2>\cdots>y_{v-1}>y_v=0.$$
...

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### 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 ...

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### 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 $P$.
...

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### 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 ...

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### 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 & \...

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### 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}...

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### 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:...

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### 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(\...

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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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### 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 $...

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### 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\...

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### Minimum Euclidean squared norm in the convex hull of points with rational coordinates

This is probably known, but I have not located a reference.
Let $P$ be the convex hull of $k$ points in $\mathbb R^n$ with rational coordinates. Consider the Euclidean square norm function $F:P\to\...

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### Is there a polytope with an essentially unique shape?

More percisely:
Question: Is there a (convex) polytope that has a unique realization up to, say, projective transformations?
I suppose I have to assume that it has more than $d+2$ vertices/facets if ...

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### Two questions on the permutohedron

The $n$-dimensional permutohedron $P_n$ is the polytope given by the convex hull of all the possible permutations of the vector $(1,2,\dots,n+1)\in\mathbb{R}^{n+1}$. So it has $(n+1)!$ vertexes.
I ...

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### 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?

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### Is there more than one pseudo-Catalan solid?

This question was asked on MSE a year ago. Motivation for this question can be found in other MSE questions here, here or here.
Convex solids can have all sorts of symmetries:
the platonic solids are ...

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### An example in symplectic geometry

$\DeclareMathOperator\SU{SU}$Let $M$ be a coadjoint orbit of dimension 6 of $\SU(3)$, and let $T$ be the maximal torus in $\SU(3)$. If we denote $\mu : M \longrightarrow \mathbb{R}^2$ the moment map ...

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### 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 $O$?

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### Edges in the convex hull of the union of random polygons

Let $P$ and $Q$ be two convex polygons in $\mathbb{R}^2$. Given $a > 0$, denote by $aP$ its image under the dilation by $a$ centered around the origin (i.e. the polygon obtained by replacing each ...

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### Conditions on $a_i,b_i$ for $i =1,\ldots,n$, so that $\arg\max_{\|x\| \le 1}\min\{a_i^\top x + b_i\mid i = 1,\ldots,n\} \subseteq \{\|x\| < 1\}$

Let $A$ be an $n \times m$ matrix with rows $ a_1,\dots,a_n \in \mathbb R^m$ and let $b=(b_1,\dots,b_n)$ be a vector in $\mathbb R^n$. Let $\mathbb B_m$ be the centered closed unit-ball in $\mathbb R^...

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### 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 ...

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### Mass distributions for high dimensional simplex and cross polytope

In this question, it is shown that the radial mass distribution of an $n$-cube (i.e. the probability density for the distance from a point selected uniformly from within an $n$-cube to the cube's ...

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### Exact volume calculation of a polytope is NP hard under which restrictions?

Computing the exact volume of a polytope given in half space representation seems to be NP-hard. One paper I found proved it is hard for rational coefficients. (However, the paper itself was behind a ...

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### How to find the dimension of the polar cone of a convex cone generated by some given vectors

Suppose we have access to a generating set $\{v_1, ..., v_k\}\subseteq\mathbb{R}^n$ of the convex cone $C=cone(v_1, ..., v_k)$, where $cone(\cdot)$ is the conical hull (i.e. nonnegative span) of ...

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### How to correctly state Cauchy's rigidity theorem?

Cauchy's rigidity theorem is usually cites briefly as
Any two (convex, 3-dimensional) polyhedra with pairwise congruent faces are themselves congruent.
As a more formal generalization to general ...

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### The area of cross-section of polyhedron or polytope

Consider a polyhedron or polytope $P$ in $\mathbb{R}^n$. Let $F$ be a bounded face of $P$. Now we shift and rotate this polyhedron such that $F$ lies on the hyperplane $\{t=0\}$, and $P\subseteq\...

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### Which polytopes can be deformed while keeping their edge-lengths?

Let $P\subset\Bbb R^d$ be a convex polytope (a convex hull of finitely many points). Lets call it flexible, if it can be continuously deformed while
keeping its combinatorial type, and
keeping its ...

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### Is there a non-orthogonal linear deformation of a polytope that preserves edge-lengths and vertex-origin-distances?

Is there a polytope $P\subset\Bbb R^d$ (convex hull of finitely many points, not contained in a proper affine subspace), and a linear, but non-orthogonal transformation $T\in\mathrm{GL}(\Bbb R^d)\...

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### 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 $...

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### 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 ...

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### Solid angles at points in an orthosimplex

Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...

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### Can every simple polytope be inscribed in a sphere?

It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that
all vertices end up on a common sphere, and
the ...

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### Distance to the “boundary” of a polyhedral complex

Suppose I have a polyhedral complex $\{P_1, \ldots, P_k\}$ and let $S := \cup_{i = 1}^k P_i$. I am interested in a function which measures the distance from a point $x \in S$ to the "boundary&...

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### 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 ...

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### Curve with no embedding in a toric surface

I am looking for a smooth proper curve $C$ such that there does not exist any closed embedding $C \to S$ where $S$ is a (normal projective) toric surface.
Since $C$ is smooth I believe it suffices to ...

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### Separating a lattice simplex from a lattice polytope

Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...

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### Probability of landing inside the convex hull of previously sampled points

Let $\{X_i\}_{0\leq i\leq\infty}$ be i.i.d. random vectors in $\mathbb{R^d}$.
I would like to show that the probability of one point being in the convex hull of the others goes to one with the number ...

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### Source request: Optimal bounds on signings of points from a convex body

I recently came across an old survey of problems in discrete geometry: https://pdfs.semanticscholar.org/c350/f4d4a9466fa6708d99ec1187c63d89bed20f.pdf
Problem 2.1 from the list caught my eye. It states ...

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### Polytope with most faces

Fix $m,n \in \mathbb{N}$ with $m \ge n+1$. Take $m$ points in general position in $\mathbb{R}^n$ and let $P$ be their convex hull. What is the maximal number of (external, codimension-one) faces that $...

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### Is the following generalization of the Caratheodory theorem true?

The colorful Carathéodory theorem (Bárány, 1982) considers $d+1$ "colors" $X_1,\ldots,X_{d+1}\subseteq \mathbb{R}^d$, and a point $x$ in the convex hull of each color ($x\in \text{conv}(X_i)$...