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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Maximal edge length of symmetric polytopes
For me, a polytope is the convex hull of finitely many points. It is said to be vertex-transitive / edge-transitive if its symmetry group acts transitively on its vertices / edges. Let's call a ...
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Can any simplex shadow-project to a regular simplex?
Every triangle $A$ can be oriented in $\mathbb{R}^3$
so that its orthogonal projection (shadow) onto the $xy$-plane is an
equilateral triangle $Q$:
&...
- 151k
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Cutting convex sets
Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference.
Can one cut every bounded convex set of the Euclidean plane into an arbitrary number $...
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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 $\...
- 13.2k
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Higher dimensional scutoids?
The recent discovery of scutoids in biological structures is fascinating. Two scutoids are depicted below (from Scientists Have Discovered an Entirely New Shape, And It Was Hiding in Your Cells), each ...
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Integer decomposition of dilated integral polytopes
For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.
Question: Given $d > n + 2$ is it true that
$$ ...
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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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Polygon of convex arcs
Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume ...
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Is there a simple test to determine whether a polytope is integral?
It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely ...
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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 ...
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Classification of space-filling (by identical copies) convex polyhedra in R^3
Is the classification of space-filling (by identical copies) convex polyhedra in R^3 is known ?
There are only 5 "parallelohedra" - filling by translation.
But if relax that property to ...
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A source for $01$-polytopes
Can you recommend any books or survey articles on $01$-polytopes, thats is, polytopes with vertices in $\{0,1\}^n$?
I am less interested in random $01$-polytopes, but more in the combinatorial ...
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Weighted area of a Voronoi cell
Let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = [0,1]\times[0,1]$, and let $w = \{w_1,\dots,w_n\}$ denote a set of weights corresponding to the $n$ points in $X$. ...
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On decomposition of polytopes
Given $m$ number of convex polytopes each with $v$ vertices and described by $h$ hyperplane inequalities in $\mathbb R^t$ are there operations on these polytopes that combine then to give an $v^{\...
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What is the lower bound for the number of facets that a general convex $d$-polytope with $n$ vertices can have?
I am familiar with Barnette's Lower Bound Theorem on the number of facets a $d$-dimensional simplicial convex polytope with $n$ vertices can have. Is there a similar result for a general (i.e. not ...
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Regularity of John's ellipsoid
Consider a finite dimensional real Banach space $E$, with norm say $|\cdot|$. Let $N$ denote the set of all norms on $E$. Suppose that $\varphi_1, \varphi_2 \in N$ have unit balls $B_1$ and $B_2$, ...
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Understanding the geometry of $H_{n}=\{\vec{x} \in [-N,N]^n:\sum_{i=1}^n x_i = 0\}$
I am not an expert in convex geometry but if we define $a_i \sim \mathcal{U}([-N,N])$ where $[-N,N] \subset \mathbb{R}$ and $S_n = \sum_{i=1}^n a_i$ I suspect that for arbitrary $N \in [1, \infty) $:
...
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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,...
- 1,813
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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}\}
...
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Complexity of 2D-Minkowski sum of non-convex polygons
I have read that the complexity of computing the Minkowski-Sum of $2$ non-convex polygons (through convex decomposition) is $O(m^2 n^2)$, where $m$ and $n$ is the number of vertices of each polygon. ...
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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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Triangulations of polytopes and tilings of zonotopes
Consider a set $A = \{ a_1,a_2,\ldots, a_n \} $ of vectors in $\mathbb{R}^d$, which lie in a common affine hyperplane. Two convex polytopes may be obtained from $A$, namely the convex hull of the ...
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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 ...
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Minuscule weights of parabolic sub-root systems are not far from dominant
Let $\Phi$ be a crystallographic root system in an $n$-dimensional Euclidean vector space $(V,\langle\cdot,\cdot\rangle)$. For a root $\alpha\in \Phi$ we use $\alpha^\vee := \frac{2}{\langle \alpha,\...
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Narayana polynomials as numerators of Ehrhart series rational functions?
The Narayana polynomials (OEIS A001263) are the h-polynomials of the associahedra (the Stasheff polytopes) and their dual simplicial polytopes (cf. the Fomin and Reading ref in the OEIS entry).
Are ...
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The intersection of two $l_1$ balls
Let $B_1$ and $B_2$ be two balls in $\mathbb{R}^n$ with respect to the $l_1$ norm that have different radii and different centers. Is there an upper bound for the number of vertices that $B_1\cap B_2$...
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How to know if convex-hull of a set contains zero?
Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha_1, \cdots, \alpha_1}_{\lambda_1}, \cdots , \...
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The number of Hamiltonian circuits on a convex polytope embedded in $\mathbb{R}^N$
Recently I wondered whether there might be a natural topological complexity measure for convex polytopes embedded in $\mathbb{R}^N$. After some reflection it occurred to me that the number of distinct ...
- 3,871
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Convex caps with prescribed edges
Let $P$ be a convex polygon in the plane $R^2=R^2\times \{0\}$, and $E$ be the edge graph of some subdivision of $P$ into convex polygons, which is $3$-connected. Does there exist a convex polyhedral ...
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4D Duoprisms based on nonconvex polygons
A duoprism is a polytope
that can be expressed as the Cartesian product of two polytopes (each of dimension $\ge 2$).
Four-dimensional duoprisms in particular have been studied:
$$P \times Q = \{ (...
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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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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\...
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Upper bound on number of vertices in intersection (and union) of simplices
Let $S_1, \dots, S_k \subset \mathbb{R}^n$ be a set of (non-regular) simplices. Let $m_i$ indicate the number of vertices of simplex $S_i$ (we do not assume it is equal to $n-1$).
Is there a simple ...
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Do plane projections determine a convex polytope?
Suppose a compact convex body $P \subset \Bbb R^3$ has only polygonal orthogonal projections onto a plane. Does this imply that $P$ is a convex polytope?
This question occurred to me when I was ...
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Continuity of the combinatorial structure of a polytope with respect to face variables
Suppose we are given a convex polytope in terms of the face variables.
That is, let $Y = (1,x_1,\dots,x_n)$ and suppose we have vectors $W_a$ in $\mathbb{R}^{n+1}$ such that the locus $W_a \cdot Y \...
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Tensor Product of Convex Sets?
I was wondering if such a concept was used anywhere. What I was thinking of is this. Consider two vectors spaces $V,W$ and convex sets $C_1 \subseteq V$ and $C_2 \subseteq W$ if we define $C_1 \otimes ...
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An integrality question about expressing an integer as a product of numbers below $n$
Let $n\ge 2$ be a natural number. Suppose that $N$ is a natural number, composed only of primes below $n$, and that can be expressed as
$$
N= \prod_{j=1}^{n} j^{x_j}
$$
where $x_1$, $\ldots$, $x_n$...
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Gaussian mean width of normal random cones
Suppose $1 \leq n < m < \infty$ are integers. For $g \sim \mathcal N(0, I_n)$ define the gaussian mean width of a non-empty set $T \subseteq \mathbb R^n$ by
$$
w(T) := \mathbb E \sup_{x \in T} \...
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Is a polytope with vertices on a sphere and all edges of same length already rigid?
Let's say $P\subset\Bbb R^d$ is some convex polytope with the following two properties:
all vertices are on a common sphere.
all edges are of the same length.
I suspect that such a polytope is ...
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Separability of Minkowski Sum of well-behaved sets
Let $A$ and $C$ be non-empty simply connected and connected subsets of $\mathbb{R}^k$ and suppose that $C$ is convex. Then is the Minkowski sum $A+C$ separable?
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Product of two matrices of convex combinations [closed]
How to show that product of two matrices $\mathbf{A}$ and $\mathbf{B}$ of convex combinations as $\mathbf{C=A*B}$ is also a matrix of convex combinations.
Convex combinations: entries of each column ...
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convex polytopes with many faces and edges but few cells and vertices
For a convex polytope $P$ in $\mathbb R^4$, denote by $N_0,N_1,N_2,N_3$ respectively the number of vertices, edges, faces, cells. By Euler's formula, we know $N_0+N_2=N_1+N_3$, which means there is a ...
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Intersection between a line and an n-dimensional parallelotope
Suppose that I have a line in an $n$-dimensional space described by
$$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$
here $A$ is known and I want to find all the possible vectors $B$ ...
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On the proportion of simplicial $d$-polytopes on $n$-vertices
I have a question regarding estimates for the proportion of simplicial $d$-polytopes on $n$-vertices.
Let $c_s(n,d)$ denote the number of combinatorial types of simplicial $d$-polytopes on $n$ ...
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Conjecture on tilting modules for an Auslander algebra
On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism
classes of modules, occurring as the $i$-th summand of ...
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On two different descriptions of Delzant polytopes
I have seen two different ways of describing a Delzant polytope:
From Canna Da Silva https://people.math.ethz.ch/~acannas/Papers/toric.pdf, a Delzant polytope is a polytope in $\mathbb{R}^{n*}$ ...
- 1,227
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Realization spaces of 3-dimensional polytopes with fixed face areas
It is a well-know result (Steinitz, 1922) that the realization space of 3-dimensional convex polytopes with fixed combinatorics is contractible.
A proof of this theorem can be found for instance in ...
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Anything similar to cone product formula (for convex polytopes)?
The convex polytope flag vector ring $\mathcal{R}$ satisfies the cone product formula
$$
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: ...
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Convex hull with genus information
Are there convexity generalizations that admit genus information?
For example in genus $1$ is there a way to think of this polyhedron as convex while this polyhedron as non-convex? Any two points can ...
- 13.9k
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Database of polyhedra
As part of many hobbies (origami, sculpting, construction toys) I often find myself building polyhedra from regular polygons. I am intimately familiar with all of the Archimedean and Platonic solids, ...
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