4
votes
2answers
147 views

Build a topological polytope with a specified CW-structure

I am a topologist and not quite familiar with the tools for building a polytope. I would like to build some topological polytope which is an somewhere in between permutohedron and associahedron which ...
2
votes
1answer
75 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 ...
1
vote
0answers
51 views

Does the set of points minimizing their distance to a multiset of convex polytopes result in a polytope?

Let $\mathbb{R}^n$ be a normed affine space of finite dimension $n$, and $d : \mathbb{R}^n \times \mathbb{R}^n \mapsto \mathbb{R}^+$ be the distance derived from the norm under consideration. A convex ...
4
votes
1answer
137 views

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 ...
2
votes
1answer
85 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 $\Delta$ ...
1
vote
1answer
207 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
283 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 ...
3
votes
1answer
268 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
175 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, ...
2
votes
0answers
203 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 ...