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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 ...
user1420303's user avatar
1 vote
0 answers
109 views

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 $\Delta_n$, ...
Xin Nie's user avatar
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11 votes
0 answers
323 views

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 ...
M. Winter's user avatar
  • 12.8k
8 votes
0 answers
135 views

How many simplicial spheres with $n$ vertices and $N$ facets?

Let $s_d(n,N)$ be the number of different $d$-dimensional simplicial spheres on $n$ labelled vertices and $N$ facets (= $d$-simplices). I am in search for the best know upper bounds, especially for $d\...
M. Winter's user avatar
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3 votes
0 answers
91 views

Minkowski problem for polytopes: the origin of necessary condition

Minkowski's uniqueness theorem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets. Theorem (Minkowski). Let $A_i$ be positive faces areas ...
Alexey Ustinov's user avatar
3 votes
0 answers
96 views

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 ...
Edoardo Rizzi's user avatar
9 votes
2 answers
567 views

Book for matroid polytopes

I have made a study of polytopes with the books of Ziegler and "Integer Programming" of Conforti, my main goal is to study matroid polytopes; to study matroids I have thought about the book &...
Wrloord's user avatar
  • 229
2 votes
0 answers
99 views

Alternative construction for the loop space (?)

There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\...
Andrea Marino's user avatar
5 votes
1 answer
91 views

How to bulge out the curved edges of the stereographic tesseract?

You probably already saw such a representation of the tesseract: I did something similar on my blog for the truncated tesseract: The vertices in 3D are the stereographic projections of the original ...
Stéphane Laurent's user avatar
1 vote
0 answers
41 views

About the number of faces of the conification of a polytope

Let $P\subset\mathbb{R}^n$ be a polytope of dimension $(n-1)$ such that the origin $\vec{0}\not\in\text{Aff}(P)$, where $\text{Aff}(P)$ denotes the affine hull of $P$ in $\mathbb{R}^n$. Now, we ...
ElliptCg's user avatar
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2 votes
0 answers
86 views

Is it possible to deduce Poincaré duality from duality of polytopes?

I'm having trouble understanding Poincaré duality, as it seems unmotivated. Here for instance: https://math.stackexchange.com/a/14469/454016 Poincaré duality is explained through a duality of ...
Alexander Praehauser's user avatar
1 vote
0 answers
41 views

Parametrize regions of positivity of a polynomial

I realize that this problem is extremely generic, so I am pessimistic that there may be concrete solutions, but let me try... Consider a multi-variate polynomial $P(x)$, is it possible to find ...
giulio bullsaver's user avatar
4 votes
0 answers
96 views

Is there a 5-cell-600-cell honeycomb?

Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
Daniel Sebald's user avatar
2 votes
1 answer
57 views

Counting the number of pair of d-uplets with upper bounded distance

Consider two d-uplets $u = (u_1,...,u_d)$ and $v = (v_1, ..., v_d)$ both living in $\mathbb{N}^d$ with $d$ a positive integer. They both verify $$(*) \sum_{i=1}^d u_i = \sum_{i=1}^d v_i = k$$ with $k$ ...
Ludwich's user avatar
  • 55
4 votes
0 answers
216 views

Characterization of curves contained in the boundary of convex bodies

Given a continuous closed curve $\gamma$ in $\mathbb R^n$ does there exist a convex body $K$ (convex set with non-empty interior) such that $\gamma\subset \partial K$? I am looking for a reference to ...
Vadim Semenov's user avatar
1 vote
0 answers
43 views

Barnes-Wall lattices’ contact polytopes

The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?
Daniel Sebald's user avatar
1 vote
1 answer
118 views

Are simplicial polytopes a dense set?

Consider the space of non-empty, compact, and convex subsets of $\mathbb{R}^d$ equipped with the Hausdorff metric. Are simplicial polytopes a dense subset of that space? Probably this is just a ...
user avatar
4 votes
2 answers
256 views

A rational polytope that is not a 01-polytope?

A 01-polytope is the convex hull of some points $S\subseteq\{0,1\}^n$. I wonder, which polytopes can be represented (combinatorially) as 01-polytopes? There are polytopes that cannot have rational ...
M. Winter's user avatar
  • 12.8k
4 votes
1 answer
112 views

Given a polytope $P$ with bipartite edge-graph, if the bipartition classes are equal in size and lie on spheres, is $P$ inscribed?

Suppose that $P\subset\Bbb R^n, n\ge 3$ is a (full-dimensional) convex polytope with a bipartite edge-graph $G=(V_1\cup V_2,E)$ (for example, a zonotope). Suppose further that there are concentric ...
M. Winter's user avatar
  • 12.8k
2 votes
0 answers
45 views

Can every non-hemi uniform polytope tile hyperbolic space?

Can every uniform polytope which is not a hemipolytope tile hyperbolic space on its own?
Daniel Sebald's user avatar
1 vote
0 answers
56 views

Are cells of 4-polytopes a convex polyhedron by definition?

I'm going by the Wikipedia definition for a 4-polytope. Do by definition, cells of 4-polytopes have to be a convex polyhedra? If not, then are there polyhedra with non-convex faces? If yes, is it the ...
Ron Michal's user avatar
0 votes
0 answers
98 views

Explicit equation for border of the Minkowski sum of sets

Assume we have sets of the form $$ M_j = \{x\in\mathbb{R}^d : f_j(x) \le 0,x \ge 0\} $$ where $x\ge 0$ means $x_i \ge 0 \quad \forall i=1,\dots, d$. Goal I am looking for an (explicit) representation ...
Felix B.'s user avatar
  • 367
5 votes
1 answer
168 views

Looking for clarification of C-H Sah's definition of abstract scissors congruence

In C-H Sah's book Hilbert's third problem: scissors congruence, the author defines the data for abstract scissors congruence in order to prove Zylev's theorem by combinatorial means in great ...
Roselyn van Lauwe's user avatar
6 votes
1 answer
245 views

Realizing spherical complexes as convex polytope

A spherical polytope is the intersection of some closed hemispheres which is non-empty and does not contain a pair of antipodal points. A spherical complex is a tiling of the whole (d−1)-dimensional ...
hyyyyy's user avatar
  • 285
0 votes
0 answers
83 views

Realizability of abstract polytopes

What are the conditions that allow an abstract polytope to have a non-skew (but not necessarily convex) realization in Euclidean space?
Daniel Sebald's user avatar
2 votes
0 answers
35 views

Finding uniform polytopes in 3D versus in 4D

How was the exhaustive uniform polyhedron search (which found the great dirhombicosidodecahedron) conducted? What’s so difficult about doing the same thing but in four dimensions?
Daniel Sebald's user avatar
4 votes
1 answer
69 views

Simplicial polytope with regular cones

Let $P$ be a convex simplicial polytope in $\mathbb{R}^n$. Can we find a convex simplicial polytope $P_0$ in $\mathbb{R}^n$ combinatorially equivalent to $P$, satisfying the following condition: The ...
user73577's user avatar
  • 405
0 votes
0 answers
95 views

Possible new convex uniform polytope

Does there exist a convex uniform 9-polytope obtained by diminishing the 9-hypercube, removing 480 of its 512 vertices and turning each 8-hypercube facet into an 8-orthoplex?
Daniel Sebald's user avatar
4 votes
1 answer
382 views

Triangulation of a simplex

I am looking for a triangulation of an $n$-dimensional simplex such that all sub-simplices are of comparable size, and are "as close as possible" to a regular simplex : the latter property ...
Bruno's user avatar
  • 43
2 votes
0 answers
90 views

Dodecahedron deformation II

(Follow-up to this question) Can a dodecahedron be deformed into a great stellated dodecahedron while maintaining the number of dimensions each element occupies?
Daniel Sebald's user avatar
6 votes
1 answer
255 views

Can a dodecahedron be deformed into a great stellated dodecahedron?

Can a convex regular dodecahedron be deformed into a great stellated dodecahedron while keeping all pentagons planar and all edges of nonzero length the whole time?
Daniel Sebald's user avatar
1 vote
1 answer
101 views

Graph diameter of the omnitruncated $E_8$ polytope

What is the graph diameter of the 1-skeleton of the omnitruncate of the $E_8$ family of uniform 8-polytopes?
Daniel Sebald's user avatar
3 votes
0 answers
115 views

Expanded 24-simplices in the Leech polytope

The vertex coordinate set for the contact polytope of the Leech lattice listed on Wikipedia contains all permutations of: $\{4,-4,0^{22}\}$ $\{-3,1^{23}\}$ $\{3,-1^{23}\}$ The convex hull of these ...
Daniel Sebald's user avatar
3 votes
0 answers
222 views

Textbooks/References for Solid Angles? [closed]

Are there any good textbooks that consider the properties of solid angles for polytopes? Being not the most well-versed in geometry, I am unsure of where to start looking. Thank you very much!
rianko's user avatar
  • 31
1 vote
0 answers
48 views

Lattice deformations of regular polytopes

It is trivial to see that the 24-cell, all hypercubes, and all polytopes with simplicial facets, can be deformed into lattice polytopes, and this blog post implies the same is true for the ...
Daniel Sebald's user avatar
1 vote
0 answers
71 views

Diminishing of the $4_{21}$

One of the projections of the $4_{21}$ polytope (https://en.m.wikipedia.org/wiki/4_21_polytope) into four dimensions positions its vertices as those of two concentric 600-cells scaled by the golden ...
Daniel Sebald's user avatar
2 votes
0 answers
54 views

Inverting "codimension matrix" for polytopes?

Let $P$ be an abstract polytope. Let's construct its square matrix $A$ as follows. Its lines and columns are labelled by all faces of $P$, of all dimensions. Put $A(F_1,F_2)=t^m$ if $F_1$ is a subface ...
Dasha Poliakova's user avatar
2 votes
1 answer
111 views

Quasiconformal map from a subset of $\mathbb{C}$ to a polytope

Question. Does a quasiconformal map exist between a subset of $\mathbb{C}$ (such as a unit disc or rectangle) and a polytope? Here, I take a polytope to be a two-dimensional surface that could be ...
Talmsmen's user avatar
  • 547
0 votes
0 answers
70 views

Regiment map from Coxeter-Dynkin diagram

The following problem arises in the attempt to enumerate uniform polytopes: Given a Wythoffian polytope, what are all other Wythoffian polytopes which use maximal subsets of its vertices and edges, ...
Daniel Sebald's user avatar
2 votes
0 answers
92 views

An easy way to recognize the edges of an orbit polytope?

Given a finite (orthogonal) matrix group $\Gamma\subseteq\mathrm O(\Bbb R^d)$ and a point $x\in\Bbb R^d$. The corresponding orbit polytope is $$\mathrm{Orb}(\Gamma,x):=\mathrm{conv}\{Tx\mid T\in \...
M. Winter's user avatar
  • 12.8k
0 votes
0 answers
92 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 ...
Turbo's user avatar
  • 13.7k
4 votes
1 answer
284 views

Intrinsic definition of a cone in a normal fan

Let $P\subseteq \mathbb{R}^n$ be a full dimensional polytope. Let us assume that $P$ has a facet description with the following inequalities: $$ \left<x,u_F\right> \geq -a_F$$ where $u_F\in \...
Luis Ferroni's user avatar
  • 1,889
3 votes
0 answers
56 views

Classifying/enumerating vertex-transitive simplicial polytopes

I'm interested in understanding the class of simplicial polytopes in $\mathbb R^n$ whose Euclidean isometry group $G$ acts transitively on the vertices. These are examples that I know of: simplicial ...
Brent Kerby's user avatar
5 votes
3 answers
564 views

Convex lattice polygons with equal area and perimeter

A convex polygon all of whose vertices have integer coordinates is a convex lattice polygon. Do there exist mutually non-congruent convex lattice polygons which have the same area and same perimeter? ...
Nandakumar R's user avatar
  • 5,765
4 votes
1 answer
145 views

A combinatorial characterization of the central inversion of a polytope?

Given a convex full-dimensional polytope $P\subset\Bbb R^d$ (convex hull of finitely many points and not contained in any proper affine subspace) and a symmetry thereof (a linear map $\smash{T\in\...
M. Winter's user avatar
  • 12.8k
1 vote
0 answers
69 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). ...
Daniel Sebald's user avatar
3 votes
0 answers
39 views

Are there uniform compounds of 135 $BC_8$ polytopes?

The Coxeter group $D_8$ is an index-135 subgroup of $E_8$. One of the consequences of this is that the rectified 8-orthoplex, whose coordinates can be given as all permutations and sign changes of $\{...
Daniel Sebald's user avatar
3 votes
0 answers
103 views

Are there any other regular compounds?

Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected ...
Daniel Sebald's user avatar
6 votes
1 answer
534 views

What are the "simplest" polytopes with an automorphism group of $\mathrm M_{11} \hspace {-1.25pt} $?

Do any polytopes have an automorphism group of the smallest of the sporadic groups, the Matthieu group $\mathrm M_{11} \hspace {-1pt} $? Indeed, they must exist. What are the simplest such polytopes ...
OzoneNerd's user avatar
  • 179
5 votes
1 answer
228 views

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 ...
M. Winter's user avatar
  • 12.8k