All Questions
24 questions
2
votes
1
answer
147
views
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 ...
- 13.6k
13
votes
0
answers
378
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 ...
- 13.6k
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 ...
- 131
4
votes
0
answers
223
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 ...
1
vote
0
answers
47
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?
- 2,773
4
votes
1
answer
117
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 ...
- 13.6k
1
vote
0
answers
58
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 ...
- 11
6
votes
1
answer
277
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 ...
- 305
0
votes
0
answers
97
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?
- 2,773
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 ...
- 2,773
5
votes
1
answer
264
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 ...
- 13.6k
3
votes
1
answer
106
views
Solid angles at points in an orthosimplex
Given a point ${\bf x} = (x_1,x_2,\dots,x_n)$ in the orthosimplex $K = \{(x_1,x_2,\dots,x_n)\ : \ 0 \leq x_1 \leq x_2 \leq \dots \leq x_n \leq 1\}$, what proportion of a ball of radius $\epsilon$ ...
- 19.7k
19
votes
1
answer
928
views
Can every simple polytope be inscribed in a sphere?
It is known that not every convex polytope (even polyhedron, e.g. this one) can be made inscribed, that is, we cannot always move its vertices so that
all vertices end up on a common sphere, and
the ...
- 13.6k
4
votes
3
answers
347
views
Minimal data required to determine a convex polytope
Let $P\subset \Bbb R^d$ be a convex polytope.
Suppose that I know
its combinatorial type (aka. the face-lattice),
the length $\ell_i$ of each edge, and
the distance $r_i$ of each vertex from the ...
- 13.6k
4
votes
0
answers
114
views
Can we combine the symmetries of two polytopes to create a more symmetric polytope?
Suppose that there are two combinatorially equivalent (convex) polytopes $P_1,P_2\subset\Bbb R^d$, that is, both with the same face lattice $\mathcal L$.
The symmetry group $\mathrm{Aut}(P_i)\subset\...
- 13.6k
6
votes
1
answer
212
views
A polytope with congruent facets and an insphere that is not facet-transitive?
Is there a $d$-dimensional convex polytope (convex hull of finitely many points, not contained in a proper subspace), with $d\ge 4$ and the following properties?
All facets are congruent,
it has an ...
- 13.6k
9
votes
2
answers
321
views
Is a polytope that has in-spheres for faces of all dimensions already regular?
Let $P\subset\Bbb R^d$ be a convex polytope (convex hull of finitely many points).
A $k$-in-sphere of $P$ is a sphere centered at the origin to which each $k$-face of $P$ is tangent. So a 0-in-sphere ...
- 13.6k
5
votes
1
answer
361
views
What is known about the duals of cyclic polytopes?
What is known about the duals of cyclic polytopes, in particular, their facets (or equivalently, the vertex-figures of cyclic polytopes)?
In even dimensions, all facets of the dual are ...
- 13.6k
3
votes
1
answer
111
views
Reference for "every 5-dimensional polytope has a 3-gonal or 4-gonal face"
It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.
Alternatively, I would be ...
- 13.6k
5
votes
2
answers
304
views
Is there a 4-polytope without 3-gonal and 4-gonal faces, other than the 120-cell?
The question is in the title:
Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?
I consider only convex polytopes (convex ...
- 13.6k
2
votes
0
answers
103
views
Polytopes with large dihedral angles
The regular $d$-simplex has dihedral angle $\arccos(1/d)<90^\circ$, and the $d$-cube has dihedral angle exactly $90^\circ$.
The maximal dihedral angle of a prism over a $(d-1)$-simplex is also $90^\...
- 13.6k
3
votes
1
answer
152
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
- 13.6k
8
votes
3
answers
390
views
Is there any edge- but not vertex-transitive polytope in $d\ge 4$ dimensions?
I consider convex polytopes $P\subset\Bbb R^d$. The polytope is called vertex- resp. edge-transitive, if any vertex resp. edge can be mapped to any other by a symmetry of the polytope.
I am looking ...
- 13.6k
2
votes
1
answer
141
views
Volume of a polytope with relaxed constraints
Consider a polytope in $n$ dimensions defined by a set of linear constraints:
$$P = \{x \in \mathbb{R}^n : Ax \leq b\}$$
where A is some $m \times n$ constraint matrix, and $b = (b_1,\ldots,b_m)$ is ...
- 21