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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Integral hull of a polyhedron Q is polyhedron
Let $Q \subseteq R^n$ be a rational polyhedron and let $Q_I=Convexhull(Q \cap Z^n)$. By finite basis theorem, we have $Q=P+C$ for some rational polytope $P$ and finitely generated cone $C$ where $C=R_+...
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Growth polynomial of the Associahedron graph ? (Is it approximately Gaussian ?)
Consider Associahedron, consider graph build from its vertices and edges. Choose some vertex. Let us count the number of vertices on distances $k$ from the selected vertex. Write a generating ...
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Meaning of the Ehrhart polynomial at $-1/2$?
I am studying a large collection of lattice polytopes, all of them being simple and empty. The dimension can be any integer. The dilatation by $2$ gives non-empty polytopes.
For many of these ...
- 3,587
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Which polytopes have compact realization spaces?
Let $P\subset\Bbb R^d$ be a convex polytope.
Its reduced realization space is the space of all combinatorially equivalent polytopes modulo projective transformations.
I am interested in polytopes for ...
- 13.6k
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Tiling with one of each 3D shape
Encouraged by the positive solutions to my question,
Tiling with one of each shape,
I'd like to pose the $\mathbb{R}^3$ equivalent:
Q. Is there a tiling of $\mathbb{R}^3$ by (bounded) polyhedra, one ...
- 151k
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Is a ball the hardest body to approximate by polytopes (in the Banach–Mazur metric)?
$\DeclareMathOperator\conv{conv}\DeclareMathOperator\Vol{Vol}$In the paper "An extremal property of the hypersphere" by Macbeath, the following functionals were introduced (here $n$ is fixed,...
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How to calculate the vertices of a convex polytope (k-DOP)
I am currently reading Christer Ericson's Real-Time Collision Detection Book. The topic I'm particularly interested in, is the chapter about Discrete-orientation Polytopes (k-DOPs). In his words "...
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A palindromic formula for simple convex polytopes
Let $P$ be a simple convex $d$-polytope (a $d$-dimensional convex polytope in which the number of edges incident to a vertex is $d$) and let $n_i$ be the number of $i$-faces of $P$. Is it true that ...
- 161
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Lattice points in the boundary of a Minkowski sum of two convex lattice polygons
Let $P$ and $Q$ be two convex lattice polygons in $\mathbb{R}_+^2$ and let $P+Q$ be their Minkowski sum. Given a set $S \subset \mathbb{R}^2$, we let $L(S) =\#( S \bigcap \mathbb{Z}^2)$.
The equality $...
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Existence of a sequence of $-1/1$-polytopes with certain geometric properties
Let $P_n \subset \mathbb{R}^n$ be a sequence of polytopes (A polytope is the convex hull of finitely many points). Let $B_n \subset \mathbb{R}^{n}$ denote the Euclidean unit ball. I am interested in ...
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Shortest loop through vertices of a convex polytope
Let $P$ be a convex polytope in Euclidean space $\mathbf{R}^3$ and $\Gamma$ be a closed curve which passes through all vertices of $P$. How small can the length $L$ of $\Gamma$ be? More specifically, ...
- 7,194
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Constructing a minimum-volume outer approximation polytope with fewer facets
I am tackling the following problem:
Given a set of points $D \in \mathbb{R}^d$ and their convex hull, represented with $n$ facets, I want to construct a convex polytope $P$ with at most $m<n$ ...
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Degree of reflectional symmetry of (unbounded) convex polyhedra in Euclidean spaces
Let $U \subset \mathbb{R}^m$ be an open domain. I'm trying to come up with a measure of its degree of reflectional symmetry and I have a question. The post in two-part, where in PART I I introduce the ...
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Is the face lattice of the cube a polytope graph?
The face lattice of a
convex polytope $P\subset\Bbb R^d$ is the partially ordered set whose elements are the faces of $P$ ordered by inclusion. We can turn it into a graph by considering its Hasse ...
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Are there polytopes with precisely two realizations?
A convex polytope is projectively unique if it has a unique realization up to projective transformations. Such polytopes are somewhat mysterious but still well-studied. Examples are simplices, the ...
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Description of the generalized permutahedron
According to Postnikov, we know that the generalized permutahedron are describe as "polytopes obtained by moving vertices of the usual permutohedron so that directions of all edges are preserved&...
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What's the number of facets of a $d$-dimensional cyclic polytope?
A face of a convex polytope $P$ is defined as
$P$ itself, or
a subset of $P$ of the form $P\cap h$, where $h$ is a hyperplane such that $P$ is fully contained in one of the closed half-spaces ...
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The realization space of non-convex polyhedra - What is known?
The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (...
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A generalized permutohedron as the sum of the dilatations of the faces of the standard simplex
I am trying to understand the proof of the statement, specifically it refers to a theorem stated by Postnikov in his text on permutohedra. So, this sentence claims the following:
If $\{Y_I \}$ is a ...
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Is a convex polyhedron determined by its edge lengths and angular defects?
Let's consider 3-dimensional convex polyhedra $P\subset\Bbb R^3$.
The angular defect at a vertex $v$ is $2\pi$ minus the sum of the interior angles of the incident faces at $v$.
Question:
Is a ...
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Is there any software package to find the vertices of convex polytope where the inequality constraints are bounded by variable?
I know of this package lcon2vert that computes vertices from given inequality and equality constraints describing a bounded polyhedron. Here the bounds of constraints only accept numerical values, i.e....
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Minimizer of forward and reverse Kullback-Leibler divergence with sum constraints on marginals
Consider minimization of the Kullback Leibler divergence between two discrete distributions $p$ and $q$:
\begin{align*}
D_{KL} \left( p \parallel q \right) = \sum_i p_i \log \left( \frac{p_i}{q_i} \...
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A continuous version of Carathéodory's convex hull theorem
A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
- 7,194
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Software for computing polytopes
As can be inferred from the title, I want to do some computation on the facets representation of the polytopes given the vertices. My advisor recommended me Polymake, which is indeed useful even with ...
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Alexandrov's uniqueness theorem in Minkowski spacetime
Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$.
Each face of $P$ comes with induced metric tensor,
if the face is space-like, then it is euclidean metric;
every time-like face is isometric ...
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Probability of being inside a convex n-dimensional polytop
I am currently conducting some post-grad research about wireless transmissions with uncertain transmission delays.
As part of the research, each individual transmission is modelled using a probability ...
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Centroid of Minkowski sum
Let $A$ and $B$ be two compact convex subsets of $\mathbb{R}^n, n\geq 2$. Assume $x_A$ and $x_B$ are their respective centroid. If we form the Minkowski sum $C=A+B = \{x+y\mid x\in A, y\in B\}$, what ...
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Convex sets via fixed point equations
I have an equation of the general form
$$ X = S \cup T X $$
where $S \subset \mathbb R^n$ is a convex polytope (given by its bounding hyperplanes), $T\colon \mathbb R^n \to \mathbb R^n$ is a linear ...
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Polyhedra volume, faces and edges from vertices
Given a set of vertices in 3D corresponding to a convex polyhedron, what is the most efficient way to find its volume, faces, and edges?
I've found some techniques using convex hulls. But I think I ...
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Homeomorphism between interiors of simplex and permutohedron
The $n$-dimensional permutohedron $P_n$ is a polytope whose facets (i.e.\ codimension $1$ faces) are in 1-to-1 correspondence with all faces (of codimension${}\geq 1$) of the $n$-simplex $\Delta_n$, ...
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Zero-one pairings between sets of vectors
Let
$A\subseteq V$ and
$B\subseteq V^\star$
be spanning sets in
a finite-dimensional real vector space $V$ and
its dual $V^\star$.
Suppose that
$$
\langle b,a\rangle\in\lbrace0,1\rbrace
$$
for all
$a\...
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Can you perturb an inscribed polytope so all its edges grow?
Consider the family of convex simplicial polytopes with vertices in the unit sphere of $\mathbb{R}^n$ which have the origin as an interior point.
My question is the following:
Let $P, P'$ be two non-...
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Is the volume of the image of the moment map an invariant of a symplectic toric manifold?
Given a symplectic toric manifold is known that the image of the moment map is a Delzant polytope that fully characterises the manifold. What about the volume of the polytope? Is it possible that two ...
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Group or semi-ring on polytope family
I want to determine how to provide structure to the polytope family; I guess the main article where this is discussed is the one by Peter Mcmullen entitled “The polytope Algebra”, here they talk about ...
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Using Ehrhart polynomials to count primes?
As indicated below, one could use the Ehrhart polynomials of the simplex in number theory.
Here are the questions without context first:
Questions:
The sum $$\sum_{k=0}^t (-1)^k ( \operatorname{...
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Implementation of Friedman's algorithm of reconstructing simple polytopes
In Finding a Simple Polytope from Its Graph in Polynomial Time, Friedman gave a polynomial time algorithm on reconstructing a simple polytope from its graph. Has this algorithm been actually ...
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"Minimal" connected matroids
I'm interested in connected matroids $M$ on the ground set $[n]$ for which there is no connected matroid on $[n]$ of the same rank but with a strictly smaller set of bases (by inclusion). Equivalently,...
- 3,513
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Sampling uniformly from the convex cone
Let $n$ vectors of dimension $d$ (e.g., $n = 100$, $d = 10000$), each with infinity norm of $1$, be given. The conic combination of those $n$ vectors generates a convex cone.
How to uniformly sample ...
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Polyhedra inscribed in a sphere with mutually non-congruent, equal area faces
Two constrained versions of the main question given in this post: Polyhedrons with mutually non-congruent faces, all of equal area. An earlier post that could be related: Cutting a spherical surface ...
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Density of the set of convex polygons in the Banach-Mazur distance
Is the set of convex polygons dense in the set of convex domains in $\mathbb{R}^2$, for the Banach-Mazur distance?
Any insight for a negative or positive answer is very much welcome!
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Denominators of rational polytopes in terms of hyperplane coefficients
Let $\mathcal{P}$ be a convex polytope in $\mathbb{R}^n$ given in the form $\mathcal{P} = \{ x \in \mathbb{R}^n\colon A x\leq b \}$. Suppose that the entries of $A$ and $b$ are integers. Then it is ...
- 24.2k
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Does Kalai's $3^d$ conjecture hold for simplicial spheres?
Kalai's $3^d$ conjecture asserts that every centrally symmetric $d$-polytope has at least $3^d$ non-empty faces. This is open in general, but has been proven for simplicial polytopes.
Question: Does ...
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Conic hull of a rectangle
I have a simple question that appeared in research: For a rectangle $S :=[a_1,b_1] \times[a_2,b_2] \times \dots \times [a_n,b_n] \subset \mathbb{R}^n$. Let $p_0 = (a_1,a_2,\dots,a_n)$, and define $p_i ...
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Does there exist a finite-volume hyperbolic Coxeter polytope with these properties?
I searched for a finite-volume, hyperbolic Coxeter polytope of dimension $n \geq 4$ with the following properties $a$ and $b$.
$a$) It has exactly one ideal vertex;
$b$) if a bounded facet and an ...
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Permutohedron and triangulation of cube via Eulerian numbers
The $h$-vector of the (simplicial complex given by the boundary of the polytope dual to the) permutohedron is the sequence of Eulerian numbers $A(n,k)=\#\{w\in S_n\colon \mathrm{des}(w)=k\}$.
Example: ...
- 24.2k
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How to sample uniformly over a polytope knowing its vertex presentation?
Say that a convex polytope $P$ is presented as $P = \mathrm{Conv}(v_1, \dots , v_m)$.
I would like to sample over $P$, without generating the facet presentation of the polytope.
How can I do that?
I ...
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7
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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 ...
- 13.6k
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Eulerian polynomial from Bruhat interval - h* of something?
Let $\sigma \in S_n$ be a fixed permutation.
Consider the polynomial
$$
P_{\sigma}(t) = \sum_{\substack{\pi \in S_n \\ \pi \leq \sigma}} t^{\textrm{des}(\pi)}
$$
where $\leq$ denotes Bruhat order, and ...
- 15.8k
4
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2
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217
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On faces of polytopes
$\newcommand\ext{\operatorname{ext}}\newcommand\R{\mathbb R}$Let $A$ be a convex polytope in $\R^n$ with nonempty interior.
Consider the closed convex cone
$$K_A:=\{(l,t)\in(\R^n)'\times\R\colon\, l(x)...
- 128k
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14
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Open problems in Euclidean geometry?
What are some (research level) open problems in Euclidean geometry ?
(Edit: I ask just out of curiosity, to understand how -and if- nowadays this is not a "dead" field yet)
I should clarify a bit ...