All Questions
Tagged with computer-science mg.metric-geometry
23 questions
3
votes
0
answers
85
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Computational complexity of exact computation of the doubling dimension
Given a finite metric space $X$, the doubling constant of $X$ is the smallest integer $k$ such that any ball of arbitrary radius $r$ can be covered by at most $k$ balls of radius $r/2$. The doubling ...
0
votes
1
answer
75
views
Robustness of doubling dimension to small perturbations
Let $M$ be a metric space. Then the doubling dimension of $M$, denoted $\dim M$, is defined to be the minimum value $k$ such that every ball in $M$ of radius $r$ can be covered by at most $2^k$ balls ...
6
votes
0
answers
176
views
Approximating a ray with an integer lattice point
Take $X$ uniform on the unit sphere in $\mathbb{R}^n.$ For $r>0$, take $S_r=\{x\in \mathbb{Z}^n: \sum_i x_i^2 \leq r^2\}.$
With $\|\cdot \|$ the 2-norm, what is the distribution (or at least the ...
3
votes
2
answers
1k
views
Place N points in a 3d cube in a way that maximizes the minimum of their pairwise distances
Place $N$ points in a 3d cube in a way that maximizes the minimum of their pairwise distances.
The problem can easily be solved for $N\lt5$, but how to proceed for larger $N$?
- 41
17
votes
2
answers
2k
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Efficiently determine if convex hull contains the unit ball
Given a set of $n$ points in $\mathbb{R}^d$, is there an algorithm to determine if the convex hull contains the unit ball centered at the origin in polynomial time (in both $n$ and $d$)? The convex ...
- 3,377
7
votes
1
answer
548
views
Constructing Metrics for specific Topological Spaces, and Refinements of the Cantor-Space in particular
I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very ...
- 798
2
votes
0
answers
246
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A primal-dual (double) circle packing (coin graph) question
I know that any 3-connected simple planar graph with a designated outside face (outer face) has a primal-dual (double) circle packing (Brightwell-Scheinerman Theorem).
Q1- But I am not sure whether ...
- 415
5
votes
2
answers
277
views
What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?
The question is mostly in the title:
What is the smallest diameter ring a non-convex polyhedron can pass through in 3-space?
Imagine I have some non-convex polyhedron $P$, and I would like to ...
- 329
3
votes
2
answers
517
views
Threading pinholes in the wall of cylinder to pass through an internal coordinate
Imagine I take a sheet of paper and use a pin to generate an $N$x$M$ rectangular array of small holes. I then fold the sheet to form a cylinder of radius $r_c$ and height $h_c$, where there are $N$ ...
1
vote
1
answer
575
views
Decomposing a sphere (or defomed sphere) into a vertex-transitive graph with fixed-length curved edges
Please see the original problem specification (which Joseph O'Rourke was responding to in his answer) below.
Motivation -
I'm interested in a particular case of the problem where one wants to ...
1
vote
1
answer
283
views
Constructing a graph that approximates a sphere using rotationally symmetric building blocks with equal numbers of edges
I'd like to construct a graph that approximates a sphere in 3-space, but I'm placed under the following constraints:
(1) - I am only allowed to use a construction block, $v_i$, consisting of a single ...
1
vote
1
answer
141
views
Inferring geometric properties of a polytope from intersection volumes of spheres at unknown coordinates on its surface
Let's say we have some polytope $P$ in 3-space (which is not necessarily convex) as well as some number of points on its surface, $(g_1, ..., g_N)$. We are provided no information about the ...
- 309
3
votes
5
answers
813
views
Is the following two-dimensional graph likely to be globally rigid?
Consider the two-dimensional non-planar graph $G$, with known topology and edge lengths $(r_1, r_2, ... r_N) \in R$, but unknown vertex coordinates. We further specify that:
All vertices within a ...
- 309
5
votes
2
answers
629
views
Approximate search space on a 5x5x5 cube with 3 different possible classes?
Hey all,
I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...
- 153
15
votes
3
answers
9k
views
$n$-dimensional Voronoi diagram
I need to compute the Voronoi diagram of a set of points in $R^n$.
I'm quite unschooled on the topic, could someone point me to the right references so that I can
a) understand the theory behind it;
b)...
- 253
3
votes
3
answers
390
views
Can we uniquely define a graph to have the topology of a polytope via proper edge length selection?
I'll ask you to consider a situation wherein one has a series of edges for a graph, $(e_1, e_2, ..., e_N) \in E$, each with a specifiable length $(l_1, l_2, ..., l_N) \in L$, and the goal is to insure ...
- 33
5
votes
3
answers
2k
views
Optimal packing of spheres tangent to a central sphere
Please consider a central, ordinary 2-sphere $S_1$, of some radius $r_1$, and a second ordinary sphere, $S_2$, of radius $r_2$, where $r_2 \leq r_1$.
My question concerns optimal values for the ...
8
votes
1
answer
2k
views
Expected number of steps for a discrete random walk to visit every point on an N-dimensional rectangular lattice
Please imagine a discrete random walk on an N-dimensional rectangular lattice with dimensional lengths $(l_1, ..., l_N) \in L$ and total lattice points $P = \prod{l_i}$, for $i = 1, ..., N$. At each ...
- 599
99
votes
7
answers
20k
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Can we cover the unit square by these rectangles?
The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1.
It is easy to show that
$$\sum_{1 \...
- 5,502
1
vote
0
answers
578
views
Cluster-preserving and distance-maximizing embedding into Hamming Space?
I have a set of data, each instance in the real $[0,1]^{d}$. However, it's actually all in a relatively small range around 0.5, clustered into classes in even smaller ranges. The actual origin of the ...
- 2,383
3
votes
4
answers
3k
views
Finding the union of N random circles arbitrarily (or conspiratorially) placed on a two-dimensional surface
Please consider a two-dimensional surface populated with a set of Cartesian coordinates $(x_i, y_i)$ for $N$ circles with individual radii $r_i$, where $r_{\min} < r_i < r_{\max}$.
Here, the ...
- 599
1
vote
2
answers
1k
views
Calculating the surface area distribution of two-dimensional projections for a polytope
My question concerns the existence of a nice (deterministic?) method/algorithm for calculating the distribution of surface areas for two-dimensional projections of an arbitrary polytope (or convex ...
- 599
9
votes
4
answers
838
views
Characterizing a tumbling convex polytope from the surface areas of its two-dimensional projections
My general question concerns what we can learn about an arbitrary, three-dimensional convex polytope (or convex hull of an arbitrary polytope) strictly from the surface areas of its two-dimensional ...