All Questions
4,827 questions
9
votes
2
answers
534
views
A question about the dispersion points of connected metric spaces
Let $C$ be an infinite, separable and connected metric space. If $C$ becomes totally disconnected when one of its points $p\in C$ is removed, does every closed ball of $C$ with
positive radius and ...
- 6,215
4
votes
4
answers
588
views
Cosine sum problem
Consider any n points on the circumference of a circle. Draw a straight line link between each pair of points with a link weight consisting of the cosine of the angle the link subtends at the centre.
...
- 41
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
8
votes
1
answer
359
views
Can any rectangle be inscribed in any convex figure?
Can any rectangle be inscribed in any convex figure?
- 2,302
6
votes
1
answer
500
views
Can any triangle be inscribed in any convex figure?
Can any triangle be inscribed in any convex figure? i.e. given a convex figure and a triangle can we transpose and scale and rotate that triangle so that its vertices are on the boundary of the ...
- 2,302
8
votes
1
answer
700
views
Upper bound for tetrahedron packing?
There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there ...
- 7,857
3
votes
2
answers
956
views
Simple definition of the Hausdorff measure using squared paper
I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...
- 20.2k
-2
votes
1
answer
840
views
Generic coordinate system representations [closed]
Please excuse the verboseness which follows, as the question is rather basic, so I would like to state it carefully so that it will not be accidentally neglected as automatically trivial. If, after my ...
2
votes
2
answers
640
views
Sorting a binary matrix diagonal in polynomial time while preserving rows
Is there a polynomial time solution to sort an arbitrary binary square matrix in polynomial time by rows so that the diagonal contains a 1 if any row contains a 1 in that column?
For example given ...
- 121
0
votes
1
answer
1k
views
For Ax = b, x and b unknown vectors, how do I solve the x that maximizes min(b_i)?
Given a matrix $A$, each element $A_{i,j} \geq 0$, find the vector $\vec x$ that maximizes the minimum element in $\vec b$ ($\vec b = A \vec x$). Note that this is not a linear equation system as I ...
- 135
19
votes
2
answers
1k
views
Are there space filling curves for the Hilbert cube?
There is a surjective continuous map $[0;1]\rightarrow [0;1]^2$ ("space filling curve"). Using such a map one can easily get space filling curves for all finite dimensional cubes.
So my question is: ...
- 11.1k
8
votes
1
answer
2k
views
Point cloud that maximizes the minimum pairwise distance in Euclidean space
I am interested finding the collection of points in the Euclidean space that has the maximal minimal pairwise distance subject to an average norm constraint, that is, how to maximize
$min_{i \neq j} |...
- 1,839
5
votes
1
answer
796
views
Minimizing variance of distances between points when mean distance is fixed
In Rd, I have n > d+1 points. The mean distance between pairs of points is 1. How can I minimize the variance of the distances (equivalently, the mean squared distance)? I'm mainly interested in d &...
- 3,612
6
votes
1
answer
715
views
Elementary problem about triangles inside a convex polygon
Let P be a convex polygon with area A(P), and to each side of P, attach the largest area triangle possible that lies entirely within P. Must the sum S(P) of the areas of these triangles always satisfy ...
- 1,318
33
votes
3
answers
2k
views
Polar body of a convex body that avoids a lattice
Let $K \subset {\bf R}^d$ be a symmetric convex body (an open bounded convex neighbourhood of the origin with $K = -K$) with the property that $K + {\bf Z}^d \neq {\bf R}^d$, i.e. the projection of $K$...
- 114k
2
votes
1
answer
342
views
Straight line on the Poincare disk hitting points almost everywhere
Consider the tiling of the Poincare disk $\mathbb{D}$ by identified octagons (i.e., representing a torus with genus 2). Suppose inside each octagon is a subset A such that the octagon minus A is a ...
- 21
7
votes
2
answers
1k
views
G-spaces and manifolds
In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
The space is metric
The space is finitely compact, i.e., a ...
- 579
1
vote
1
answer
492
views
Сomplete homogeneous space which is not locally compact
It is well-known theorem that every locally compact, homogeneous, metric space is complete.
Does anybody know example of complete, homogeneous, metric space which is not locally compact?
- 141
14
votes
2
answers
1k
views
Polygonal billards programs
I'm looking for software that will give billiard trajectories in arbitrary plane polygons. After much work I was able to produce this figure.
(source)
It was a good exercise, but at this point I ...
- 22.8k
99
votes
7
answers
20k
views
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
11
votes
7
answers
2k
views
Topological spaces that resemble the space of irrationals
(This question actually arose in some research on number theory.)
I once learned that any countable dense subspace of any Euclidean space $\mathbb R^n$ is homeomorphic to the rationals $\mathbb Q$.
...
- 2,858
5
votes
4
answers
954
views
literature on geometrical viewpoint on calculus of variations for physics
What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
...
- 921
18
votes
2
answers
2k
views
Assistance with understanding parent/child relationships in Pythagorean Triples
I want to start by apologising for what is probably a weak attempt at a question on a site like this, but I'm having trouble understand a concept that doesn't seem to be properly explained elsewhere - ...
- 283
2
votes
2
answers
749
views
Projective transformation between polygons.
Extending my earlier question about linear transformations, what's the easiest way to test if there exists a projective linear transformation that takes one polygon to another (in $\mathbb{R}\mathbb{P}...
- 5,901
2
votes
2
answers
1k
views
Linear transformation takes a polygon to another one.
Say we have $n$-gons $P$ and $Q$. Is there any necessary condition for $Q = f(P)$, for some linear transformation $f : \mathbb{R}^2 \to \mathbb{R}^2$?
Sorry if this is too elementary / general.
- 5,901
5
votes
2
answers
3k
views
Continuous Linear Programming: Estimating a Solution
I have a "continuous" linear programming problem that involves maximizing a linear function over a curved convex space. In typical LP problems, the convex space is a polytope, but in this case the ...
- 373
47
votes
7
answers
5k
views
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
- 11.4k
4
votes
1
answer
275
views
Symmetry of the integer gap
Are there results that bound the asymmetry of the duality gap of an integer program? That is to say, if the difference between the LP solution and the IP (primal) solution is $a$, is there a function ...
- 183
11
votes
2
answers
3k
views
Algorithm for embedding a graph with metric constraints
Suppose I have a graph $G$ with vertex set $V$, edge set $E \subseteq {V \choose 2}$, a poistive integer $d$, and a weight function $w:E \to \mathbb{R}^{+}$. Is there a nice algorithmic way to decide ...
- 7,857
3
votes
0
answers
192
views
Velcro surface: is it possible to have a surface which is everywhere infinitesimally a cone, but not a normed group?
Is there any example of this velcro-like space? Looking for a LOCALLY COMPACT COMPLETE metric space $(X,d)$ such that:
(A)-it has a metric tangent space $(T_{x}X, d^x)$ at any point $x \in X$,
(...
- 785
1
vote
1
answer
322
views
Settling a circular argument: room for one more?
By using a regular hexagonal arrangement it is simple to fit 19 identical circles into a larger circle of five times the radius with no circles overlapping. This leaves an area equal to six smaller ...
- 13
4
votes
0
answers
735
views
Theorems in affine geometry which can be proved using only dilations, generalized to metric spaces
Background: significant parts of E. Artin, Geometric algebra, Wiley-Interscience, New York, 1957 can be seen as consequences of the statement
(1) "the inverse semigroup generated by dilations in ...
- 785
7
votes
1
answer
665
views
What is the Cheeger constant of a cubical subset of the cubic lattice?
The Cheeger constant of a finite graph measures the "bottleneckedness" of the graph, and is defined as:
$$h(G) := \min\Bigg\lbrace\frac{|\partial A|}{|A|} \Bigg| A\subset V, 0<|A|\leq \frac{|V|}{2}...
- 1,916
17
votes
4
answers
2k
views
Planar sets where any line through the center of mass divides the set into two regions of equal area.
This question is influenced by the following riddle:
You are given a rectangular set in the plane with a rectangular hole cut out (in any orientation). How do you cut the region into two sets of ...
- 7,197
4
votes
4
answers
589
views
Measures of the complexity of a metric
I am seeking a measure of the "complexity" of a surface $S$,
a quantity that reflects how widely the metric varies from spot to
spot. I am primarily interested in surfaces topologically
equivalent to ...
- 151k
16
votes
6
answers
3k
views
Smallest area shape that covers all unit length curve
On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this ...
- 613
7
votes
1
answer
865
views
Computer power in plane geometry
I often hear that modern computer programs "may prove any theorem in elementary Euclidean geometry". Of course, as stated it is false - say, they can not prove theorems about $n$-gons for ...
- 109k
15
votes
1
answer
2k
views
Pythagorean theorem for right-corner hyperbolic simplices?
My answer to the "Favorite equations" question was the Pythagorean theorem for right-corner tetrahedra:
Euclidean: $A^2+B^2+C^2=D^2$
Hyperbolic: $\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}−\...
- 1,230
1
vote
1
answer
324
views
A question about dissecting spherical triangles
Do there exist spherical triangles which are not isoceles but are the union of a finite collection of
(two or more) congruent triangles with pairwise disjoint (and non-empty) interiors?
- 6,215
5
votes
2
answers
1k
views
Inequality involving probability measures [closed]
I have been working on a problem(alternate minimization) where I want to establish an inequality in which I am stuck.
An $\alpha$- parameterized version of the divergence(Kullback-Leibler) takes the ...
- 779
7
votes
1
answer
815
views
Rolling a convex body: Geodesics vs. rolling curves
What are the curves of contact on a convex body $B$ rolling down an inclined plane?
Assume $B$ is smooth, and there is sufficient friction to prevent slippage.
Certainly, one can develop a geodesic ...
- 151k
3
votes
1
answer
375
views
Connections between a polytope's symmetry group and the existence of periodic orbits
Given an $n$-dimensional convex polytope $P$, one may set into motion a point-mass, starting on one of the facets of $P$, which travels along a straight trajectory inside $P$ except on collision with ...
- 269
5
votes
0
answers
427
views
The Gömböc and monostatic objects
This made the popular news a few years ago. Summary: it's the first homogeneous, convex solid to be found with only one stable and one unstable mechanical equilibrium when resting on a flat surface. I ...
- 3,612
16
votes
4
answers
2k
views
Neusis constructions
Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis?
See http://en.wikipedia.org/wiki/Constructible_number and http://en.wikipedia....
user5810
3
votes
2
answers
301
views
Optimizing the layout of Infinite Suburbia
Infinite Suburbia is a Euclidean plane, P. All residents live in open unit disks which, like caravans, can travel around but are stationary most of the time. When stationary, these disks lie in a ...
- 3,612
32
votes
5
answers
2k
views
Nonconvex manhole covers
One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width — if ...
- 5,275
3
votes
1
answer
335
views
When is the neighbourhood of a set a ball?
In euclidean n-space, it's easy to show that given a set $S$ of radius $< r$, the $a$-neighbourhood of $S$ is a ball, for any $a \geq 2r$.
Proof: Let $S$ be contained in $B_r(y)$, $y \in \mathbb{...
- 7,800
4
votes
2
answers
3k
views
Computational geometry, tetrahedron signed volume
Short version: I'm trying to compute the orientation of a triangle on a plane, formed by the intersection of 3 edges, without explicitly computing the intersection points.
Long version: I need to ...
- 143
5
votes
2
answers
1k
views
Applications of minmax theorem(s)
Intro We suppose $X$ and $Y$ are nonempty sets and f: $X\times Y \rightarrow \mathbb{R}$. A minimax theorem is a theorem that asserts that, under certain conditions,
$$ \inf_Y \sup_X f = \sup_X \...
1
vote
3
answers
2k
views
How to solve Linear Programming problem with tighter Integer Programming constraints
I want to learn a bit about Linear Programming.
After some research, I decided to solve the Cutting Stock problem as an example to learn. After doing some more research, I feel like I finally ...
