All Questions
12 questions
7
votes
3
answers
805
views
Wrapping a convex polyhedron with string
This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P \...
- 151k
6
votes
0
answers
74
views
Roundest polyhedra: how well can we bound the edge count of their faces?
By "roundest" I mean having the lowest surface area for the highest volume, given a fixed number of faces $n$. There've been a few questions about them on here (including from me), but I'm ...
- 3,612
5
votes
0
answers
199
views
Existence of a honeycomb composed by nearly-hyperspherical $d$-dimensional cells having the same shape and size
Let $\mathcal{H}$ the class of all honeycombs composed by $d$-dimensional cells $C$ having all the same shape and size in a $d$-dimensional space $\mathcal{S}$.
Let $s(C)$ and $\ell(C)$ be ...
- 1,677
4
votes
1
answer
1k
views
Algorithm for the shortest path through all the points of a 2D cloud
I have an array of points with their coordinates X and Y. Each point represents a bus stop.
I need to sort the points in a sequence by giving them sequence numbers, so that the path from the first to ...
- 61
4
votes
1
answer
363
views
Trade-off between hypervolume and diameter of $d$-dimensional shapes having a hypercubic smallest bounding box
Given any $d$-dimensional shape $X$, let $V(X)$ be its $d$-dimensional volume, and let $\ell(X)$ be the length of the longest line segment connecting two points of $X$.
Let $\mathcal{S}_C$ be the set ...
- 1,677
4
votes
0
answers
246
views
Distance properties of the permutations of a set of points in a Euclidean space
We are given a set of $n$ distinct points $S=\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_n\}$ in a Euclidean space $\mathbb{R}^d$, where the distance between two points $\mathbf{x}_i,\mathbf{x}_j\...
- 1,677
3
votes
1
answer
143
views
Combinatorial Euclidean geometry problem
Let $\mathcal{S}^d_{\epsilon}$ be the collection of all sets $S:=\{\mathbf{x}_1, \mathbf{x}_2, \ldots \mathbf{x}_{d+1}\}$ of $d+1$ points in a $d$-dimensional Euclidean space such that, for a given ...
- 1,677
2
votes
2
answers
163
views
References for geometric properties of optimal Euclidean traveling salesman tour
Consider a finite set of points $V \subseteq \mathbb{R}^2 $ as a TSP-instance under the standard $\| \cdot \|_2$ norm. (TSP stands for traveling salesman tour.) We know that every optimal TSP tour $T$ ...
- 29
2
votes
0
answers
131
views
Maximum number of regions in a disk partitioned by pairs of parallel chords
We are given a disk $D$ in $\mathbb{R}^2$. Let $C$ be its boundary (i.e., the circle bounding $D$ on its plane). Let $P(n,d)$ be a set of $n$ pairs of chords of $C$ such that for each $\{c,c'\}\in P(n,...
- 1,677
1
vote
0
answers
67
views
Conjecture on the increasing efficiency of the shortest minimum-link polygonal chains covering any grids of the form $\{0,1,2\}^k$ as $k$ grows
From the well-known Nine dots problem, we know that we need a polygonal chain with at least $4$ edges to connect the $9$ points of the planar grid $G_{3,2}:=\{\{0, 1, 2\} \times \{0, 1, 2\}\} \subset \...
- 1,451
1
vote
0
answers
68
views
Facility location and traveling salesman
This question is based on Distributing points evenly on a sphere and Facility location on manifolds
The 'dispersal problem' (which can be mapped to packing disks in many cases) places $n$ points in a ...
- 5,979
0
votes
0
answers
176
views
How to find a configuration of lines
In $\mathbb{R}^3$, can anyone help find a configuration of 5 lines such that the minimum of the smallest semi-axis lengths of the ellipsoid $ \mathbf{x}^T \mathbf{A} \mathbf{x} = 1 $, where $\mathbf{A}...
- 61