All Questions
28 questions
19
votes
1
answer
448
views
Precise estimate for probability an $n$-point set has diameter smaller than $1$
This question was inspired by an earlier question that I answered but would like a more precise bound for.
Consider random points $x_1, \dots, x_n$ in the unit ball in $\mathbb R^d$, uniformly and ...
- 148k
17
votes
4
answers
823
views
Sweep-segment bot: Will this random walk sweep the plane?
This model is inspired by the random behavior of the
Roomba sweeping robot.
Let a unit segment $ab$ in the plane be placed
initially with $a=(0,0)$ and $b=(1,0)$.
The segment is first rotated a ...
- 151k
16
votes
3
answers
2k
views
A random walk on random lines
I am wondering if this random walk remains finite with positive probability.
Start with three lines $A,B,C$ that are extensions of an equilateral triangle.
Let $p_0$ be one corner. Generate a line $...
- 151k
16
votes
1
answer
1k
views
Random polycube shapes
I am wondering if it is hopeless to obtain any firm results
on the following model of a "random polycube shape."
First, a polycube in $\mathbb{R}^3$
is a connected face-to-face gluing of unit cubes.
(...
- 151k
15
votes
2
answers
571
views
Spearing rolling hula hoops
Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, d-\...
- 151k
15
votes
1
answer
2k
views
Ping-pong relief map of a given function z=f(x,y)
I have an idea to design a type of
Galton's Board
to "draw" a relief map of a given two-dimensional function $z=f(x,y)$.
A typical Galton's Board drops, say, ping-pong balls through a series
of evenly ...
- 151k
10
votes
1
answer
673
views
A random variation on Pólya's orchard problem
Pólya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, e....
- 151k
9
votes
4
answers
371
views
Diameter of random segment intersection graph?
I have an even number of points $n$ randomly distributed (uniformly) in a disk.
Then the points are randomly connected to form $n/2$ segments, a perfect
matching.
Finally, I form the intersection ...
- 151k
9
votes
1
answer
338
views
Visibility in a growing orchard
This is a variant on Polya's orchard problem.1,2
Suppose trees are planted randomly in the plane.
The question is: How many trees are visible from the origin as
their radii grow?
More precisely, ...
- 151k
7
votes
1
answer
318
views
Finding a short path using $(0.99n)!$ permutations
Suppose I have $n$ points $x_1,\dots,x_n$ that are all independent uniform samples in the unit square, and I'd like to find a short path (in terms of Euclidean length) that touches all of them (a ...
- 4,049
7
votes
3
answers
377
views
Expected minimum face angle of random convex polyhedron in $\mathbb{R}^3$
Let $P_n$ be a "random convex polyhedron" in $\mathbb{R}^3$ of $n$ vertices, where "random" could follow any one
of a number of models:
(1) the convex hull of $n$ points randomly and uniformly ...
- 151k
7
votes
0
answers
162
views
Approximating any convex shape in $\mathbb{R}^d$ with a polytope having $\mathrm{poly}(d)$ facets
We denote by $V(A)$ the $d$-volume of any convex set $A$. Furthermore, given any two convex sets $A,B\in\mathbb{R}^d$, we denote by $V_{A,B}$ the $d$-volume of the symmetric difference $V\left(A \...
- 1,677
7
votes
0
answers
122
views
Discrepancy of the finite approximation of the Lebesgue measure
Let $\mu$ be a probabilistic measure on the unit square $Q$ which is the average of $N$ delta-measures in some points in this square; let $\lambda$ denote the Lebesgue measure on $Q$. What is the rate ...
- 109k
7
votes
0
answers
209
views
Stabbing disks in space, or: Galactic alignment
I have a collection of $n$ unit-radius disks in $\mathbb{R}^3$, whose centers are
random within a sphere of radius $R>1$, and which are each oriented randomly.
I'd like to find a line $L$ that ...
- 151k
5
votes
1
answer
430
views
Volume of a shape whose boundary consists of portions of spheres symmetrically placed about the origin in $d\gg 1$ dimensions
We are given a convex shape $S$ in the $d$-dimensional Euclidean space, whose boundary is formed by portions of $2d$ different spheres, one portion per sphere. The radius of each sphere is the same, $...
- 1,677
4
votes
1
answer
263
views
Knotted TSP tours in 3D?
In the plane, the Euclidean TSP tour never crosses itself—it is always a simple polygon.
I am wondering if there is a similar constraint for the Euclidean TSP tour
of points in $\mathbb{R}^3$.
...
- 151k
4
votes
0
answers
144
views
Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
- 1,677
4
votes
0
answers
94
views
Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius
I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...
- 161
4
votes
0
answers
173
views
On understanding Discrete-Valued Stochastic Processes( time series, panel data )
It seems to me that a significant proportion of work in probability theory, statistics and machine learning are on understanding continuous-valued, relatively weakly dependent, or linear dependent ...
- 41
3
votes
1
answer
473
views
On 4 random points in a rectangle [closed]
Given a bounded rectangular area, I generate 4 random points. What is the probability that the fourth point lie within a triangle formed the first 3?
How would I attack this problem? The goal is to ...
- 133
3
votes
1
answer
197
views
Three-dimensional Apollonian spirals
Given mutually (externally) tangent spheres $S_1$, $S_2$, $S_3$, $S_4$, let $S_n$ be the unique sphere externally tangent to $S_{n-1}$, $S_{n-2}$, $S_{n-3}$, and $S_{n-4}$ for $n \geq 5$.
Let $P_{\...
- 19.7k
3
votes
1
answer
218
views
Bounding the number of facets of a polytope to approximate a given convex shape in higher dimensions
We are given a convex shape $S$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume $V(S)$ of $S$ be $\tfrac12$ (I guess nothing changes for any other fixed ...
- 1,677
3
votes
0
answers
187
views
Approximating any $d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having $\mathrm{poly}(d)$ facets
Given any convex set $A\in\mathbb{R}^d$, we denote by $V(A)$ its $d$-volume. Furthermore, given any two convex sets $A_1,A_2\in\mathbb{R}^d$, we denote by $V_{A_1,A_2}$ the $d$-volume of the symmetric ...
- 1,677
2
votes
1
answer
151
views
Given an input point in $\mathbb{R}^n$, select (one of) the closest point(s) from a fixed large set of points given in advance
We are given a set $S$ of $m\gg 1$ points in $\mathbb{R}^n$.
In the problem I am trying to solve, in a sequential fashion, we obtain a new point $p_r\not\in S$ at each round $r\ge 1$ and the goal is ...
- 1,677
2
votes
1
answer
190
views
Estimating the volume of a convex shape in higher dimensions based only on normal sections
We are given a $d$-dimensional convex shape $S$ inscribed in the hypercube $[-1,1]^d$. We want find an approximation of its volume based only on a set of curves given by the intersection of the $S$ ...
- 1,677
2
votes
1
answer
404
views
Euclidean distance bound with geometric constraints
Let $S_n$ be a set of $n$ points belonging to $\mathcal{B}_d:=\{\mathbf{x}\in\mathbb{R}^d:\|\mathbf{x}\|_2\le 1\}$, where $d\ll \log(n)$.
Let $s_n$ and $\ell_n$ be respectively defined as follows:
$$...
- 1,677
2
votes
1
answer
110
views
A questions concerning Laguerre/Voronoi tessellations
Fix $n>1$ distinguished points $x_1,\ldots, x_n\in \mathbb R^d$, the Voronoi tessellations are the subsets $V_1,\ldots V_n\subset\mathbb R^d$ defined by
$$V_k~~ := ~~ \big\{x\in\mathbb R^d:\quad |...
user128095
1
vote
0
answers
153
views
Is there a polynomial expression for the volume of the following set?
Denote the unit $\ell_2$ ball in $\mathbb{R}^n$ as $\mathcal{B}_n$. It is widely kown that for a convex body $\mathcal{K}\subseteq \mathbb{R}^n$, the $n$-dimensional volume of the parallel body $\...
- 550