All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
4
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0
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153
views
Perimeters of nested convex spherical polygons
I seek a reference—not a proof—that if $P_1$ and $P_2$
are two convex polygons on a sphere composed of geodesic segments,
contained in a hemisphere, and
$P_1 \subseteq P_2$, then the ...
- 151k
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
4
votes
0
answers
202
views
An isoperimetric inequality for "order" polytopes
I am looking for an isoperimetric inequality for order-like polytopes.
An order polytope $K\in \mathbb{R}^n$ is defined by a set of linear inequaities:
$$ \forall i \; 0\leq x_i \leq 1 $$
and
$ ...
- 243
4
votes
0
answers
443
views
Intersection of pencils in $\mathcal{R}^2$
Consider $9n$ pencils through non-collinear points $p_1, \ldots , p_{9n}$ in $R^2$ each consisting of at most $n$ concurrent lines. Define the intersection $S$ of these pencils to be the set of points ...
3
votes
2
answers
831
views
Kepler conjecture: Are there only two most efficient packings or could there be more than two?
Today I attended a talk by Terence Tao, attended by (I'm guessing) probably at least a couple of thousand people, in which among other things he said it had been proved that no packing of spheres in ...
3
votes
2
answers
202
views
Existence of lattices whose circles have bounded number of points
For any plane lattice $\Lambda= \{ mA+nB: m,n \in \mathbb Z \}$, with $A,B$ linearly independent vectors in $\mathbb R^2$, we define the set of the circles in $\Lambda$ as
$$\mathcal K(\Lambda) = \...
- 315
3
votes
3
answers
314
views
4-polytope with vertices at the binary octahedral group
Does anybody know if there is a convex polytope in $R^4$ with vertices at the binary octahedral group (identifying $H$ with $R^4$).
The binary tetrahedral group lies at the vertices of the so-called ...
- 1,147
3
votes
2
answers
323
views
Minimum weight triangulation of lattice points in a circle
Let $r$ be a natural number, and consider the $\mathbb{Z}^2$ lattice points
$S$ inside or on the circle $C$ of radius $r$ centered on the origin.
Let $P$ be the convex hull of $S$; so $P$ is inscribed ...
- 151k
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
222
views
Number of lines of symmetry of a set of lattice points
Given some finite $S\subseteq\mathbb R^2$, it is clearly possible for $S$ to have arbitrarily many lines of symmetry. However, it is not very clear if the same is necessarily true for subsets of $\...
- 1,890
3
votes
1
answer
366
views
Illumination from visible lattice points with inverse square intensity
It is well known that the number of $\mathbb{Z}^2$ lattice points visible from
the origin is $6/\pi^2$, about $61$%.
See, e.g.,
What fraction of the integer lattice can be seen from the origin?.
I am ...
- 151k
3
votes
1
answer
152
views
Triangles that can be cut into mutually congruent and non-convex polygons
It is easy to note that an equilateral triangle can be cut into 3 mutually congruent and non-convex polygons (replace the 3 lines meeting at centroid and separating out the 3 congruent quadrilaterals ...
- 5,979
3
votes
2
answers
438
views
If a polytope is centrally symmetric and combinatorially equivalent to a zonotope, is it a zonotope?
A zonotope is a polytope whose 2-faces are centrally symmetric.
Question: If a polytope $P$ is centrally symmetric and combinatorially equivalent to a zonotope, is it itself a zonotope?
- 13.6k
3
votes
1
answer
285
views
Name this kimberling center
The lines which connect the vertices of a triangle with the tangent points between the Spieker circle and the medial triangle are concurrent. Which kimberling center does this point correspond to?
- 1,109
3
votes
1
answer
253
views
Nagel line of a tetrahedron?
It's well known that there is an analogy for the Euler line in a tetrahedron, but is there also an analogy for the nagel line of a tetrahedron? I can't seem to find any decent literature talking about ...
- 1,109
3
votes
1
answer
554
views
Calculate the discrete set of points B which are in the convex hull of the set of points A
This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...
- 151
3
votes
1
answer
373
views
Radial tilings with variable area ratios
I was looking at this neat page on logarithmic spiral tilings when a question popped up:
http://www.uwgb.edu/dutchs/symmetry/log-spir.htm
It seems that in all of the tilings shown, the area of each ...
- 93
3
votes
2
answers
232
views
Partition of polygons into 'congruent sets of polygons'
Definition: Two finite sets of polygons $A$ and $B$ are congruent if we can match polygons in $A$ in a one-one manner with polygons in $B$ with each matched pair of polygons mutually congruent.
...
- 5,979
3
votes
1
answer
118
views
Question arise from kissing number in 2 dimension
I'm considering an extended problem of kissing number in $\mathbb{R}^2$.
Suppose I have a given disc $\mathcal{D}$ of radius 1/2 and infinitely many discs all of radius 1/2 and all these discs and ...
- 1,495
3
votes
1
answer
191
views
Maximal $\pi/2$-separated subset of the sphere
A subset $A$ of a metric space is called $\varepsilon$-separated if
$$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$
(Notice that the inequality in my definition is strict.)
What is the ...
- 21.8k
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
156
views
Points in general position on a small grid
A point set $P$ is said to be embedded in $\mathbf{Z}^2$ in general position, if no three points lie on a common line. Assume that $|P|=n$, I am interested in the smallest $k \times k$ integer grid in ...
- 133
3
votes
3
answers
311
views
Are there infinite sets of stellations of polyhedra?
Lists of stellations of polyhedrons are given particular rules like in the book The Fifty Nine Icosahedra which follows "Miller's Rules".
There seems to be no "correct" ruleset to use, so more ...
- 2,615
3
votes
1
answer
152
views
Are there any more polytopes whose 2-faces are identical 4-gons?
What are examples for convex polytope $P\subset \Bbb R^d,d\ge 3$ for which holds
$P$ is 2-face transitive (that is, all 2-faces are equivalent under the symmetries of $P$), and
all 2-faces of $P$ are ...
- 13.6k
3
votes
1
answer
206
views
Separating points of shifts of a finite set in the plane
Let $A\subset \mathbb{R^2}$ be a finite set such that $|A|=k^2$. Let $x_i\in \mathbb{R^2}$, $i=1,2,3,4$, be four points in the plane in general position (no three lie on any line).
Let us form the ...
- 2,288
3
votes
1
answer
295
views
Monotone polygons (and polyhedra) with respect to a point
Dear mathoverflow community,
working on a visualization project I encountered a geometric problem, which I have not yet heard about and am interested in solving algorithmically. However a mere hint ...
- 31
3
votes
1
answer
495
views
The circle with minimal radius covering known finite set of points on a plane
Given some points on a plane, how to determine the circle with minimal radius covering all these points?
- 143
3
votes
1
answer
201
views
Simplex with edges of length at least s having smallest circumradius
Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I ...
- 185
3
votes
1
answer
364
views
On distances between points on the plane
Take a set of $2n$ points in the plane and assume that no open set of diameter $1$ contains more than $n$ of these points.
Question: can we pair up the points so that the distance between the points ...
- 2,288
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
1
answer
238
views
Least area and least perimeter triangles that contain a convex planar region - how different can they be?
Is there a planar convex region whose enclosing triangles of least perimeter and least area have different areas and different perimeters? And if so, which region maximizes the difference between the ...
- 5,979
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
3
votes
1
answer
190
views
On some centers of convex regions based on partitions
These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'.
Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
- 5,979
3
votes
2
answers
179
views
Number of bitangents to convex polytopes
Let me state my question prior to defining terms:
Q. Let $P_1$ and $P_2$ be disjoint convex polytopes
in $\mathbb{R}^d$ of $n$ vertices each.
What is the maximum number of distinct bitangent
...
- 151k
3
votes
1
answer
472
views
3D discrete curves geometry: method to order points in a same "general" ordering
I have a collection of 3D discrete curves $\{C_i\}$, each with a different number of points $N_i$:
$$ C_i = [p^i_0, p^i_1, ..., p^i_{N_i}] \text{ with } p^i_k=[x^i_k, y^i_k, z^i_k] \text{ i.e. } C_i \...
- 131
3
votes
1
answer
1k
views
Bound on maximum distance between points on a unit N-Sphere
I want to select M points on the N-sphere such that $min_{i\neq j,i,j\in \{1..M\}} ||x_i - x_j||$ is maximized.
Are there good upper bounds for this max-min distance?
- 41
3
votes
1
answer
386
views
Pointers/Papers on subdivision of planar quadrilateral meshes (PQ-Mesh) in 3D?
I'm interested in the subdivision of planar quadrilateral meshes (PQ-Meshes). Meshes consisting only of planar quadrilaterals, like discrete Voss surfaces and alike. I've been searching the web
for ...
- 133
3
votes
1
answer
236
views
Non-inherited symmetries of shadows of point sets
Sometimes a point set in Euclidean space may have a shadow with an unexpected symmetry. The purpose here is to ask when this happens or when it doesn't happen (in some generality).
This requires a ...
- 1,277
3
votes
1
answer
94
views
Convex polygon shadows: Shortest equivalent segments
Let $P$ be a convex polygon.
Q1. What is the shortest collection of line segments $S$ inside $P$
with the property that both $P$ and $S$ have the same sequence of orthogonal shadows
as $P$ and $S$ ...
- 151k
3
votes
1
answer
212
views
Monotile that tiles when you apply a rubber band
My (non-mathematician) friend asked me a physics/tilings question that maybe someone here is interested in dissecting, or can point to the literature if this problem has been studied.
Does there ...
- 6,652
3
votes
1
answer
84
views
Tilings of lattice polytopes by transformations of lattice polytopes
A quasi-lattice polytope is a polytope obtained by reflections, translations, and rotations of lattice polytopes. In a tiling of a lattice polytope by quasi-lattice polytopes, are all quasi-lattice ...
- 745
3
votes
1
answer
111
views
Reference for "every 5-dimensional polytope has a 3-gonal or 4-gonal face"
It seems to be folklore that every 5-dimensional convex polytope has a 3-gonal or 4-gonal face of dimension two. I was not able to track down a source for that claim.
Alternatively, I would be ...
- 13.6k
3
votes
2
answers
344
views
Is a vertex- and edge-transitive polytope already a uniform polytope?
I want to consider (convex) polytopes $P=\mathrm{conv}\{p_1,...,p_n\}\subset\Bbb R^d$ which are both, vertex- and edge-transitive (or maybe stronger: 1-flag-transitive).
Question: Is every such ...
- 13.6k
3
votes
1
answer
190
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 ...
3
votes
1
answer
205
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)\...
- 43
3
votes
1
answer
518
views
n-dimensional Delaunay Triangulation of Lattices
I have several questions concerning the Delaunay triangulation of a high dimensional lattice.
Given an $n$-dimensional lattice $L$ and its Delaunay triangulation (partition of $R^n$ into simplices ...
- 31
3
votes
1
answer
439
views
Convex n-polytope general position vectors to general position vectors of tetrahedron
I asked this question in a comment to this question, but got no response. I thought that perhaps it needed more exposure, so I made it a question in itself.
Define a set of general position vectors $...
- 4,842
3
votes
1
answer
239
views
The realization space of non-convex polyhedra - What is known?
The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (...
- 13.6k
3
votes
1
answer
111
views
Constrained morphing of polygons
This post continues 'Constrained morphing' of planar convex regions
If an $m$-gon $P_m$ is to be morphed (altered continuously) into an $n$-gon $P_n$ with same area and perimeter, can one ...
- 5,979