# Tagged Questions

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### High-Dimensional Analogs of Polygon Spaces

[Edit: I had a mistake in the numerology (took d=6,5 instead of d=5,4). Edit: I mistakenly identified my mistake, it is 6,5 but I got the indices shifted by one.] Background: Polygon spaces Given a ...
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### Finite union of closed convex sets is triangulable?

I posted this question on Stackoverflow, but didn't get an answer. Let $A_1, \ldots, A_k \subseteq \mathbb{R}^n$ be closed convex sets. Is the union $\bigcup_{i=1}^k A_i$ triangulable, that is, ...
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### What are some triangulations of Grassmannians?

A while ago I heard that there was no known triangulation of the Grassmannian of 3-planes in 6-space. To believe a statement like that, you have to be a little bit ungenerous about what you mean by "...
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### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...
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### Comparing layered triangulations of 3-manifolds which fiber over the circle.

I am sorry but I am reposting this question because I wasn't logged in when I first asked it. Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which ...
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### What is the number of equitriangulations of the n-cube?

I wonder if this question has been considered before and if anything is known. My search attempts have failed so far. Let's consider the n-dimesnional cube, $[0,1]^n$, and let's call a simplex with ...
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### Proving the Gauss-Bonnet theorem for embedded surfaces using triangulations

I heard this really neat elementary proof of the "Gauss-Bonnet Theorem" : Let $S$ be a surface embedded in $\mathbb{R}^3$. Now take a triangulation of that surface, and approximate the surface with a ...
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### Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold. It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ? ...
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### Efficient topological triangulations of non-convex polyhedra

Does every polyhedron in $\mathbb{R}^3$ with $n$ triangular facets have a topological triangulation with complexity $O(n)$? Suppose $P$ is a non-convex polyhedron in $\mathbb{R}^3$ with $n$ ...
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### Smoothing of piecewise Euclidean Riemannian metrics

Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on ...
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### Characterizing Convex Configurations of Quadrupels of Coplanar Points via Linear (In-)equalities between Distance Sums or Differences

Given 4 points $A$, $B$, $C$ and $D$ in general position in the euclidean plane, is it possible to determine from the 6 distances $AB$, $BC$, $CD$, $AD$, $AC$ and, $BD$ alone, whether every point is a ...
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### Do random triangulation edge-flips maintain randomness?

Let $S$ be a fixed set of $n$ points in the plane in general position. Let $T$ be a triangulation of $S$, (somehow) selected uniformly at random from all triangulations of $S$. (There are an ...
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### Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (...
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### Simplex in convex polytope, pulling triangulation

Let $P$ be a convex $d$-dimensional polytope. I have two questions, related to triangulations of $P$. Question 1: Let $p$ be in the interior of $P$. Can I always find a triangulation of $P$, such ...
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### Most regular way to triangulate $\mathbb{R}^3$?

By "regular", I'm going by a property of Delaunay Triangulation, which is to maximize the minimum angle. In $\mathbb{R}^2$, tiling with equilateral triangles gives you a minimum angle of 60 degrees. ...
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### (∞,n)-category of triangulated cobordisms

What is an accepted definition of a (∞,n)-category of triangulated cobordisms? Is there one that has a forgetful functor to (Rezk - Hopkins -) Lurie's smooth cobordisms? Does it shed light on how ...
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### Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following. Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...
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### Dehn-Sommerville relations for $\Delta$-complexes

Let $M$ be a closed, triangulated manifold of dimension $m$ and $K(M)$ be its triangulation. Let $f_i$ denote the number of $i$-simplices of $K(M)$. As proved by Klee the face numbers satisfy the ...
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### Hamiltonian circuit

Let us consider a disk with one labelled point on the boundary and $n$ labelled points in the interior. Let T be a triangulation of the whole disk with vertices on the labelled points such that T ...
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### regular triangulations of the product of two simplices

Is description of all regular triangluations of $\Delta^n\times \Delta^k$ known? (Regular triangulations are those which correspond to vertices of Gelfand--Kapranov--Zelevinsky secondary polytope, or, ...
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### Euclidean triangulation of the plane with degree 7 at each vertex.

Hyperbolic plane has a beautiful triangulation by congruent hyperbolic triangles where all the vertices of the triangulation have degree 7, this is of course not possible in the euclidean plane, even ...
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### Triangulations of translation surfaces whose edges are shorter than the diameter

Let $S$ be a compact translation surface (i.e. a surface endowed with a singular flat metric such that singular points are locally isometric to a cone of angle an integer multiple of $2\pi$, and that ...
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### practical algorithm for constrained triangulation in two dimensions?

I'm looking for an algorithm that is easy to implement in practice (resulting in small amount of code), preferably incremental. As far as I know, the biggest problem with incremental constrained ...
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### mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations. Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...
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### Homotoping a 2-plane field on a closed orientable 3-manifold to a contact structure

I am a beginner in Contact Geometry. To prove that the inclusion $\text{Cont}(M)\hookrightarrow \text{Dist}(M)$ induces surjection at $\pi_0$ level, the closest I got was based on Ko Honda's notes. ...
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### Existence of triangulation of Lipschitz domains

Consider a bounded Lipschitz domain $\Omega \subset \mathbb R^n$. Q1: Can its closure $\overline\Omega$ be triangulated? Q2: If yes, can the triangulation be chosen as finite? Furthermore, how ...
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### dissections and vertices of non-convex polytopes

Let us call a finite union $P$ of $n$-dimensional compact convex polytopes in $\mathbb{R}^n$ a non-convex polytope. Recall that a dissection of $P$ is a finite collection $T$ of $n$-dimensional ...