# Questions tagged [polyhedra]

The polyhedra tag has no usage guidance.

199
questions

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### Integrality of polyhedra

Given two polyhedra in $H$ representation $P_1:Ax\leq b$ and $P_2:Bx\leq c$ which are integral are bounded when is their intersection also integral?
Given two polyhedra in $H$ representation $P_1:Ax\...

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56 views

### Number of vertices in a polyhedron

Consider polytopes
$$A_1[x_{1,1},\dots,x_{1,m_1},z_{1}]'\leq b_1$$
$$A_2[x_{2,1},\dots,x_{2,m_2},z_{2}]'\leq b_2$$
$$B[z_{1},z_{2},z]'\leq c$$
having vertex count $v_1,v_2$ and $v$ respectively.
We ...

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104 views

### Exactly counting number of vertices of a polyhedron

Suppose $Ax\leq b$ is a polyhedron, where the number of rows in $A$ is $r$, the vector $x$ lies in $\mathbb R^n$ and the rank of $A$ is $t$. Assume minimal number of hyperplane inequalities to define ...

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60 views

### Number of linear inequalities describing a polyhedron with prescribed number of vertices

If a polytope has $d$ vertices in $k$ dimensions how many linear inequalities is required to describe it?

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209 views

### Embedding an icosahedron

A transitive set in $\mathbf{R}^n$ is a finite set with a transitive group of symmetries. I want to understand how subsets of a transitive set constrain the group.
Let me start with the example of a ...

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95 views

### Distinguishable knots (with constraints) over polyhedra

I'm trying to find the number of distinguishable knot projections over certain convex regular polyhedra according to the following constraints. On each face on the polyhedron the knot will have a ...

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**1**answer

113 views

### Finding Motzkin's original paper on copositive quadratic forms

I am currently in the process of writing my thesis about copositive matrices and would like to write a chronological narrative about the ascent of these matrices to the prominent place they have today ...

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43 views

### Classification of pseudoregular polyhedra

In contrast to a regular polyhedron, which has one orbit of flags, I’ve been studying what I call pseudoregular polyhedra, which have two orbits of flags interchanged by conjugation (explained here). ...

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45 views

### 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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79 views

### Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected?

Let $\mathcal{A}_\Gamma$ be the space of convex (non-degenerate) Euclidean polyhedra with $1$-skeleton a certain polyhedral graph $\Gamma$. This space can be seen as a subset of $\mathcal{Gr}_2(\...

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117 views

### Duality argument for elliptic regularity

M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p&...

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42 views

### What are the expected values of the volumes of two classes of ellipsoids contained within the unit 3-ball, and/or what is their ratio?

Consider the class of all ellipsoids contained in the unit 3-ball, and also the subclass of those ellipsoids also contained within tetrahedra also contained in the unit 3-ball. What are the expected ...

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180 views

### 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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101 views

### What is the expected value of the volume of a tetrahedron inscribed in the unit sphere?

Four (non-coincident) points on the unit sphere determine a tetrahedron. What is the expected value of the volume of such a tetrahedron--the volume of the sphere itself being $\frac{4 \pi}{3} \approx ...

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127 views

### Clustering of vertices in an $n$-dimensional cube

Consider the vertices of an $n$-dimensional cube. The distance between two vertices is measured as the minimum number of edges between the two vertices. Now consider a subset of these vertices.
If we ...

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702 views

### 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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312 views

### Do Bernoulli polynomials know about face vectors?

This question is grounded firmly in numerology. It originates in an observation about some Bernoulli polynomials and the regular icosahedron. Let $F_{k+1}(n)=\sum_{i=1}^n i^k$ be the sum of the ...

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289 views

### Minimum number of distinct triangles for tesselating the sphere

Consider sequences of tesselations of the sphere. For instance, one such sequence might start with an icosahedron and proceed by subdividing each triangle face into 4 triangles and projecting the new ...

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108 views

### What are the known convex polyhedra with congruent faces?

Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...

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159 views

### 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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29 views

### 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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124 views

### Is Sydler's theorem concerning Dehn invariants constructive?

Sydler proved something of a converse to Dehn's negative resolution
of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that
"every two Euclidean polyhedra with the same volumes and Dehn ...

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51 views

### Find tetrahedron vertex given 3 vertices of a face and the 3 opposite angles

I have the following tetrahedron:
which I know the coordinates of $P$, $Q$ and $R$ and the value of angles $\theta_0$, $\theta_1$ and $\theta_2$.
I need to find the coordinates of vertex $E$. Is that ...

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55 views

### 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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69 views

### How to define and compute the degree of congruence of two rigid polyhedra in same type with knowing vertex coordinates?

If I have two sets of points in 3-dimensional space, each sets of points are the coordinates of vertices of a polyhedron. The two polyhedra have same type, so we don't need to consider the topological ...

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44 views

### Do continuous motions of the vertices of convex polyhedra that maintain local convexity imply global convexity? (Reference request)

A convex polyhedron has all of its internal dihedral angles in $(0, \pi)$. However, if I start with an abstract polyhedron $P$, let's say a triangulated one, so I don't have to worry about planarity ...

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49 views

### 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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66 views

### 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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44 views

### 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,...

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98 views

### 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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32 views

### 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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135 views

### Self-intersecting path of stacked regular tetrahedra

(This question occurred to me after reading
@IanAgol's reminisces
of Conway's spiral tetrahedron billiard path.)
Let $T_i$ be a regular tetrahedron,
and $P$ a collection of regular tetrahedra
glued ...

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40 views

### fast V representation update of polytope

Say that I have both the V and the H representation of a (possibly unbounded) polytope $P$. I want to append a some rows to the H representation, how can I quickly update the V representation to ...

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78 views

### Distance between two polyhedra that takes incidence structure into account

Suppose that we have two polyhedra $P_1$ and $P_2$ in $\mathbb{R}^3$. I would like to define such a metric $\rho(P_1, P_2)$ that depends on several factors, but currently I don't know how to do it ...

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138 views

### Polyhedra that can pack 3-space only in a non-vertex-to-vertex fashion

Question: Are there polyhedral units (convex or otherwise) that can pack 3D space without gaps only such that the arrangement is not vertex-to-vertex?
Same question can be asked with 'edge to edge'.
...

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140 views

### Alexandrov's rigidity in higher dimensions

If $\Phi_1,\Phi_2$ are convex polyhedra in $\mathbb{R}^3$ such that the sets of outer normals to facets coincide, but $\Phi_1$ is not a translate of $\Phi_2$, then there exist two corresponding ...

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46 views

### Essential rays in fan structure

Let $|\Sigma|$ be the underlying set of some fan $\Sigma$ in $\mathbb{R}^n$. It is well known that in general there is no coarsest fan structure on $|\Sigma|$. However, there may be some special rays ...

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49 views

### Polytopes and polyhedral cones in complex Euclidean space

Given $A \in \mathsf{M}_{m \times n}(\mathbb{R})$ and $b \in \mathbb{R}^m$, the polyhedron with respect to $A$ and $b$, denoted by $P(A,b)$, is defined by
$$ \{ x \in \mathbb{R}^n \mid Ax \le b \}.$$
...

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58 views

### Iterated polyhedron face twisting

Let $Q$ be a polygon in the plane. Modify $Q$ by rotating each edge about its
midpoint by $180^\circ$. The result is $Q$ again: No change.
This suggests exploring a similar operation in $\mathbb{R}^3$...

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4k views

### A curious relation between angles and lengths of edges of a tetrahedron

Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...

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202 views

### 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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95 views

### Products of polytopes and the normals of their facets

I need to compute the normals of the facets of certain polytopes that can be represented as products of polytopes in smaller dimensions.
Searching the bibliography I found that the facets of the ...

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**3**answers

527 views

### Alexandrov's generalization of Cauchy's rigidity theorem

Wikipedia states that A. D. Alexandrov generalized Cauchy's rigidity theorem for polyhedra to higher dimensions.
The relevant statement in the article is not linked to any source. The sources at the ...

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181 views

### Motivic strong bellows conjecture

There is a theorem due to Gaifullin--Ignashchenko stating that the Dehn invariant of any flexible polyhedron in the $n$-dimensional Euclidean space ($n\geq 3$) is constant during the flexion.
Is ...

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28 views

### When do projection maps of polyhedra factor?

Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...

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### Mixed integer formulation of union of polytopes?

Given $t$ different unbounded polyhedra $P_1:A^{(1)}x^{(1)}\leq b^{(1)},\dots,P_t:A^{(t)}x^{(t)}\leq b^{(t)}$ we are looking for the representation of $\bigcup_{i=1}^tP_i$ (not their convex hull) with ...

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331 views

### What is the average area of the shadow of a convex shape taken over all possible orientations?

If we take a sphere, its surface area is exactly 4 times the area of its shadow. How can it be generalised for any convex shape?

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123 views

### Is there a method to cut a hypercube into disjoint cubes [closed]

Since Borsuk conjecture hold for centrally symmetric convex sets in $\mathbb{R}^n$
so we can cut a hypercube into at least $n+1$ disjoint parts.
Is there a method how can one do that?

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137 views

### Can bellows make loops?

Can flexible polyhedron (hyperbolic or euclidean) have non-simply connected configuration space not containing singular polyhedra?

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108 views

### Name for facet of a cone containing all but one edge

Let $C \subseteq \mathbb R^n$ be a polyhedral cone, so generated by its edges ($1$-dimensional faces) and $F \subseteq C$ a facet (codimension $1$ face) of it containing every edge except $e$. In ...