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
918
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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 ...
- 3,301
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2
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Positivity of the coefficients of the Ehrhart polynomial of a cross-polytope
Question 35996 asks about the Ehrhart polynomial $i_d(n)$ of the
standard regular cross-polytope. It can be defined equivalently by
$$ \sum_{n\geq 0}i_d(n)x^n = \frac{(1+x)^d}{(1-x)^{d+1}}. $$
It ...
- 49.5k
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Does a triangulation without fixed simplex property always exist?
Suppose we are given a triangulable topological space $X$. If $X$ has the fixed point property (FPP), then obviously for every triangulation $K$ of $X$ and every simplicial map $f:K\to K$ a simplex $\...
- 2,074
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Covering the unit sphere by open hemispheres
Suppose $H_1,\ldots,H_{2n}$ are open hemispheres which cover $S^{n-1}$ with the property that removing any one of them leaves $S^{n-1}$ uncovered. Is it necessarily the case that the hemispheres can ...
- 219
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What is the number of equitriangulations of the n-cube?
I wonder if this question has been considered before and if anything is known. My search attempts have failed so far.
Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...
- 85.1k
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Connecting Lemma in the Alexandrov's existence theorem.
At the moment I am polishing my lecture notes which in particular cover Alexandrov's existence theorem.
Denote by $\mathbf{P}_k$ the space of isometry classes of polyhedral
metrics on the $\mathbb S^...
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Reciprocity (Ehrhart-style) for real polytopes?
Is there some sense in which the well-known Ehrhart reciprocity law for rational, convex, polytopes can be extended to any convex polytope with arbitrary real vertices?
In other words, given any ...
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3
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"Database" of simplicial polytopes/spheres
Reading through various papers on polytopes I have come across really interesting examples of simplical polytopes and non-shellable (or non-PL) simplicial spheres but sometimes it is hard to keep ...
- 263
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Detecting tilings by toric geometry
This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, ...
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Random Sampling a linearly constrained region in n-dimensions...
Hi,
So here is my problem:
Given a nonlinear, discontinous, cost function $f(x_1,x_2,..,x_N)$ along with linear constraints $x_n \ge 0, \forall n$
$x_n \le c_n$
and $\sum_{n=1}^N x_n = 1$ find an ...
- 113
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Characterizing faces of 3-dimensional polyhedra. (Related to Victor Eberhard's Theorem [1890]:)
Originally asked by Ali Dino Jumani; EXTENSIVELY EDITED by David Speyer. The previous version was a bit confused, but Steven Sivek and Graham, in the comments, figured out what was going on.
G. C. ...
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2
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Book on the tetrahedron
Does anybody know of a book containing "all you want to know about the tetrahedron"? What you want to know should include basic geometry of the tetrahedron, study of orthocentric tetrahedra, the Monge ...
- 4,382
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Determining whether a lattice is the face lattice of a polytope - NP hard or undecidable?
According to this source (p. 10), determining whether a simplicial complex is a simplicial sphere (the sphere recognition problem) is undecidable.
According to this source, determining whether a ...
- 12.6k
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Why do convex polytope options constrict with dimension, rather than expand?
There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There ...
- 149k
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How many non-equivalent sections of a regular 7-simplex?
Suppose we have a regular 7-simplex in $\mathbb{R}^8$ defined by vertices <1,0,0,...,0>, <0,1,0,..,0>,...,<0,...,0,1>. A section is a 3-dimensional linear subspace of $\mathbb{R}^8$ that ...
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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 $...
- 149k
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Tighter Caratheodory on the moment curve?
The moment curve is the set of points of the form
$$(t,t^2,t^3,...,t^n) \in R^n$$
Let $M$ be the portion of the moment curve where $t\in [0,1]$, and let $\overline{M}$ be the convex hull of $M$.
...
- 3,809
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Needle probing for a convex body
Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...
- 149k
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1
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Geometric realization of combinatorial self-duality in polytopes
Let's say I have a combinatorially self-dual polytope $P\subseteq\Bbb R^d$, i.e., its face lattice is isomorphic to its dual (you reverse the direction of the lattice order).
Question: Is it always ...
- 12.6k
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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 ...
- 12.6k
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doubly-stochastic isomorphisms of graphs
A doubly stochastic matrix that commutes with the adjacency matrix of a graph is a doubly-stochastic automorphism of that graph (definition by Tinhofer 1986). Each (classical) automorphism of a graph ...
- 3,318
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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 ...
- 12.6k
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297
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How many ways to flatten a Tesseract onto a table?
A cube can be cut and flattened out onto a table in a way that the faces stay connected and none of them overlap. There are $384$ ways to make the cuts and $11$ distinct meshes emerge (see here). And ...
- 171
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Is my open set a ball?
Let $P$ be a polytope of dimension $n$, and let $\mathcal{C}$ be a polyhedral subdivision of $P$. That means that $\mathcal{C}$ is a finite collection of polytopes whose union is $P$, such that the ...
- 2,527
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Right-angled polytopes
%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not exist ...
- 1,236
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Lower Bound on the Volume of Certain Polytopes
Given a partition $\rho\in\mathcal{P}(n)$ with $k$ blocks
$$
\rho=\{B_1,B_2,\ldots,B_{k}\}
$$
we can define the set of equations
$$
E_{i}:\sum_{j \in B_{i}}{x_{j-1}}=\sum_{j \in B_{i}}{x_j}\quad\text{...
- 3,616
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Polytopes with few vertices.
Suppose I have a convex polytope in $\mathbb{R}^d$ which I know has few vertices (in the case which prompted this question, I seem to have a polytope in $\mathbb{R}^9$ which has sixteen vertices). Is ...
- 95.6k
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When does every point in a polytope lie along a chord between its edges?
Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...
- 329
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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 ...
- 4,727
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Do maximal polyhedra have algebraic volume?
Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number?
Update: What can ...
- 6,709
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Symmetries and faces of the associahedron
The dihedral group of order $2n+2$ acts on $K_n$, the ($n-2$)-dimensional associahedron. Are there any other symmetries? References?
Does the answer to 1 change if we restrict to just the 1-...
- 1,585
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Largest regular $k$-simplex inscribed in a $d$-cube, $k < d$
The largest (by edge length) regular simplex inscribed in a unit cube
is well known in $\mathbb{R}^2$ and $\mathbb{R}^3$:
Image sources:
left: NMSU,
right: Mathworld.
A recent Amer ...
- 149k
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Number of regions formed by $n$ points in general position
Given $n$ points in $\mathbb{R}^d$ in general position, where $n\geq d+1$. For every $d$ points, form the hyperplane defined by these $d$ points. These hyperplanes cut $\mathbb{R}^d$ into several ...
- 461
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What is the convex hull of the quaternionic symmetries of the 3 dimensional cube?
It is well known that there are exactly five 3-dimensional regular convex polyhedra, known as the Platonic solids.
In 1852 the Swiss mathematician Ludwig Schlafli found that there are exactly six ...
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covering convex sets by round balls
Let $X=\{x_1,...,x_k\}\subset E^n$ be a finite subset in the Euclidean $n$-space, $r>0$ and $B(x_i,r)$ are open balls of radius $r$ centered at the points $x_i\in X, i=1,...,k$. Suppose that
$$
\...
- 31k
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Status of Barany's conjecture?
One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks:
Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$?
A convex $...
- 1,302
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Computionally efficient vertex enumeration for (convex) polytopes
Let $P \subseteq \mathbb{R}^d$ be an $\mathcal{H}$-polytope. The vertex enumeration problem asks for the set of vertices $V$ of $P$. Theoretically, the vertex enumeration problem for $P$ can be ...
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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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What's the volume of $\{x\in[0,1]^n|\sum x_i\le t\}$ for real $t$?
EDIT: Thanks for your answers and comments. There is indeed a classical easy formula, given by Pietro Majer (with a simple nice proof) in his answer below.
Given $x\in\mathbb{R}^n$, $x_i$ denotes its ...
- 1,817
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3
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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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Odds on rolling a rhombicosidodecahedron
This is more of a curiosity to me, but I'm sure I don't have the mathematical skills to answer it. That said... I took a look at several other posts with questions that relate to this one, but I ...
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Approximate volume computation and lattice point enumeration - hardness
Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.
...
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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 ...
- 882
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Given the skeleton of an inscribed polytope. If I move the vertices so that no edge increases in length, can the circumradius still get larger?
Let $P\subset \Bbb R^n$ be an inscribed convex polytope, that is, all its vertices are on a common sphere of radius $r$.
Let $G$ be the edge-graph of $P$. For convenience, assume $V(G)=\{1,\dotsc,s\}$....
- 12.6k
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2
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What is determined by the combinatorics of the shadows of a convex polyhedron?
Define the shadow of a convex polyhedron $P$ in direction $u$
to be the orthogonal projection of $P$ onto a plane whose normal is $u$.
The shadow is a convex $k$-gon.
I am wondering to what degree $P$ ...
- 149k
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2
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203
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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 ...
- 3,031
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Bi-spherical polyhedra
Bicentric polygons have been studied: a polygon all of whose vertices lie on its
circumcirle, and whose incircle is tangent to every edge:
I have not been able to find a comparable literature ...
- 149k
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3
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Minimal combinatorial data needed to define a polytope [duplicate]
Suppose I give a list of vertices $(v_1, v_2, ..., v_n)$, and a list of "adjacencies", i.e. pairs of vertices $(v_i,v_j)$. Does it exists a unique polytope that has this vertices and realises the ...
- 1,335
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Pictures of the von Neumann polytope
Are there any graphic portrayals of von Neumann polytopes in low dimensions?
- 3,850
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Positivity of Ehrhart polynomial coefficients
Are there any results stating that a given family of convex polytopes have Ehrhart polynomials with non-negative coefficients?
What methods are available for proving such a property for some family ...
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