# Tagged Questions

Euclidean, hyperbolic, discrete, convex, coarse geometry, comparisons in Riemannian geometry, symmetric spaces.

1answer
62 views

### Existence or otherwise of a set of “sufficiently intricate” open cells

In my question Existence or otherwise of a set of "sufficiently intricate" open sets, I asked about whether it is possible to partition Lebesgue-almost all of $\mathbb{R}^d$ into a finite ...
1answer
114 views

### Closest point to a dual lattice point (in particular for root lattices!)

Given a lattice $\Lambda\subset \mathbb{R}^n$ and a point $p\in\mathbb{R}^n$ outside the lattice, then I known it is a hard question to determine the set $S\subset \Lambda$ of all lattice points with ...
1answer
100 views

1answer
147 views

### How many points are in such set with the same norm-2

Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking ...
1answer
125 views

### How to show it is contained in a convex hull?

There are $(d+1)f$ points (denote the set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have  \mathcal{H}(F_i)\...
2answers
305 views

### Constructing a function over a metric space through given points

Suppose there is a compact metric space $(X,\rho)$ and a Euclidean space $\mathbb{R}^n$. There is a sequence of unequal points $\{x_1,...x_N\}$ in $X$ such that all metrics $\rho(x_i,x_j)$ are known ...
0answers
119 views

### GPS calculations under $L^p$ norms

GPS calculations require finding a sphere externally tangent to four given spheres, an Apollonian problem in $\mathbb{R}^3$. The center of that fifth sphere is one of the $16$ possible solutions to ...
1answer
202 views

### Combinatorial description of a 120-cell

I'd like a combinatorial description of the 1-skeleton of the 120-cell (roughly) along the lines of the following description of the 1-skeleton of a dodecahedron. (View all elements of product sets ...
0answers
115 views

### Connectedness of cones in the boundary of a 1-ended hyperbolic group

Let $G$ be a one-ended hyperbolic group. We can think of the boundary of $G$ as consisting of geodesic rays originating at the identity in some Cayley graph, modulo the relationship of being ...
1answer
118 views

### Self-avoiding/reflecting geodesics on a convex surface

Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$. For me $S$ is a convex polyhedron, but I am happy to view $S$ as a smooth body with positive Gaussian curvature at each point, or ...
1answer
363 views

### Limiting shape for Brillouin zones

Is it true that the limiting shape for Brillouin zones (for any lattice) is a circle? You can find the definition and the step by step construction of Brillouin zones here. This picture is taken from ...
1answer
128 views

### what's the formula of the inradius of a general simplex? [closed]

As the title, I just want to know whether there is a general formula for calculating the inradius of a n-simplex. Thank you!
5answers
1k views

### Tiling the plane with incongruent isosceles triangles

It is not difficult to tile the plane with incongruent triangles. One could tile with equilateral triangles, and then partition each equilateral into three triangles, displacing their common ...
1answer
223 views