The polytopes tag has no wiki summary.
4
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1answer
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Generalized permutahedron and random polytopes
The Birkhoff polytope $B_n$ is defined as the convex hull of the set of permutation matrices, which gives us the set of doubly stochastic matrices. A concept which is intimately related is that of the ...
5
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0answers
55 views
Labeled polytopes
In a "problem of a week" contest in my school, I gave the following problem to students: we assign to each vertex of a cube a number $1$ or $-1$. And we associate to each face the product of the ...
2
votes
1answer
79 views
Estimates on the number of vertices of reflexive polytopes
Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ ...
17
votes
1answer
500 views
An NP-hard $n$ fold integral
We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral
$$
J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} ...
1
vote
1answer
182 views
non-convex Polytope definition
I have a simple question. I read that given a vector space $N_{\mathbb{R}}$ over $\mathbb{R}$,
we can define a convex polytope in the following way:
$$P:= \Big\{ \sum_{u\in S} \mu_u u \,\Big| \, ...
5
votes
1answer
261 views
Does every simplicial polytope have a topology-preserving contractible edge?
An edge of a triangulated manifold is said to be contractible if it may be contracted to a vertex without modifying the topological type of the underlying manifold. Otherwise, the edge is ...
2
votes
2answers
295 views
Who knows this convex polytope?
I hope this is not too trivial for this forum. I was wondering if someone has come across this polytope.
You start with the rhombic dodecahedron, subdivide it into four parallellepipeds, and then ...
3
votes
1answer
255 views
moduli space of polytopes
When considering classification problems about polytopes, I sometimes has the feeling that one need to talk about certain parametrized families, i.e. moduli space of such polytopes. But neither do I ...
3
votes
1answer
168 views
Classification of lattice polytopes with small number of lattice points in the facets
Suppose $P$ is a convex lattice polytope in $Z^3$ without interior lattice points, and we require the interior lattice points of each facet(i.e. dimensional 2 faces) are neither too much nor too few, ...
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0answers
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Combinatorics- Polytopes Question [closed]
Can someone help me solve the following question please?
Let v be a vertex of a d-polytope P such that $ 0 \in intP $ .
Prove that $ P^* \cap \{ y \in \mathbb{R}^d \mid\left < y, v\right>=1\ ...
5
votes
1answer
332 views
What can be said about number-theoretic properties of the solid angle measures of polytopal cones in the weight lattice of sl(n)?
The following question might be elementary — it is too far from my area of expertise to tell. It has shown up in my research in an interesting way, which I will not go into here, but I'm happy to ...
7
votes
1answer
333 views
Building a polyhedron from areas of its faces
Is there a known algorithm which, given a finite multiset (unordered list) of integers $A$, returns a yes/no answer for ...
2
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0answers
198 views
Sampling from a partition of a hypercube by convex polytopes.
I have a binary space partitioning (BSP) tree which recursively partitions a hypercube using linear hyperplanes. That is, a hyperplane splits the hybercube in half, creating two convex polytopes. Each ...
3
votes
0answers
225 views
Polygon illumination with perturbed reflections
Here is a variation on the classical polygon illumination problem. For $c \geq 0$ we say that a mirror has reflection index $c$ if whenever a ray hits the mirror with angle of incidence $\alpha$ then ...
3
votes
1answer
208 views
Non-inherited symmetries of shadows of point sets
Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality).
This requires a ...
2
votes
3answers
540 views
An exterior angle theorem for n-dimensional polytopes?
In the plane, the exterior angle of a vertex is $\pi -$ the standard ("interior") angle, which may be negative in some cases. The following is true for non-weird polygons:
The sum of the exterior ...
9
votes
1answer
749 views
Spanning polytopes
Hamiltonian cycles (seen as spanning polygons) are interesting for several reasons (only a few of which I am aware of), but especially because not every connected graph has a Hamiltonian cycle (is ...
