All Questions
Tagged with mg.metric-geometry discrete-geometry
671 questions
9
votes
2
answers
3k
views
get a point in polygon (maximize the distance from borders)
I have several 2D polygons represented by lists of xy-coordinates of their vertices.
It is needed to get several points inside the polygon so that they lie possibly far from the polygon's borders (...
CommunityBot
- 1
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-\...
- 6,206
7
votes
2
answers
453
views
Bound on Minimal Length of Vectors in Lattice and its Dual Lattice
Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and $\Lambda^\ast$ its dual lattice. Let $d=\min_{v\in\Lambda} (v,v)$ and $d^\ast =\min_{v\in\Lambda^\ast} (v,v)$ be the minimal squared lengths of vectors ...
- 800
5
votes
0
answers
1k
views
N-balls covering n-balls
This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...
CommunityBot
- 1
11
votes
2
answers
444
views
The intersection of a circle and a rank 3 subgroup of the plane
Let $A$ be a rank 3 subgroup of the Euclidean plane, i.e. $A = \mathbb{Z} v_1 + \mathbb{Z} v_2 + \mathbb{Z} v_3$, where $v_1, v_2, v_3 \in \mathbb{R}^2$ are three $\mathbb{Q}$-linearly independent ...
- 1,371
10
votes
1
answer
535
views
Maximum number of Vertices of Hypercube covered by Ball of radius R
Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
- 6,101
20
votes
1
answer
452
views
Hidden points in polygons
Let $h(n)$ be the largest number of mutually invisible points that can be located in a
polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment
$xy$ contains a point ...
- 7,276
4
votes
1
answer
203
views
Covering a convex body with its smaller homothetic copies
Given a convex body $C\subset R^d$ and a positive real $\lambda$, any set of the form $\lambda C + x = \{ \lambda c + x \mid c\in C \}$ for some $x\in R^d$ is called a homothetic copy of $C$. The ...
- 6,210
1
vote
0
answers
60
views
Finding special vectors generated by a matrix
Let $G\in \Bbb Z^{n\times n}$ be a unimodular matrix.
Are there any efficient algorithms to find the maximum norm of a vector $v$ that satisfies $\langle\Delta(v),v\rangle=0$ over all vectors $v\in ...
- 6,210
22
votes
1
answer
663
views
Voronoi cell of lattices with the same profile
Definition 1. Given a body $V$ in $\mathbb R^n$,
the function $p_V\colon \mathbb R_+\to \mathbb R_+$
$$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$
will be called profile of $V$.
Definition 2. Define ...
CommunityBot
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0
votes
0
answers
143
views
On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)
Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...
- 6,210
3
votes
0
answers
135
views
Lattices achieving best density
Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...
- 800
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 ...
- 6,210
5
votes
1
answer
532
views
Regular lattice polygons
Suppose I want to construct an $N$-gon in the plane whose vertices are integer lattice points, and which is close to a regular $N$-gon (which means, the ratio of longest to the shortest side is within ...
- 12.3k
15
votes
1
answer
640
views
Smallest regular simplex containing the unit cube in $R^n$
What is the length $e_n$ of the edge of the smallest $n$-dimensional regular simplex $S_n$ containing the $n$-dimensional unit cube $Q_n$?
In particular, is there $n$ such that $e_n<\sqrt{2}(n+1-\...
CommunityBot
- 1
2
votes
1
answer
1k
views
Geodesic convex hulls in a graph; and their properties
This question asks for an analog of the convex hull in a graph that parallels
(as far as possible) convex sets in Euclidean space.
Let $G$ be a simple, undirected graph, and let $S \subseteq V$ be a ...
- 85.6k
6
votes
0
answers
114
views
Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces
Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...
- 161
24
votes
3
answers
1k
views
Tetrahedron insphere iteration
I know that iterating the following incircle construction approaches an equilateral triangle in the limit:
Starting with any triangle $T$, one forms $T'$ by connecting ...
- 24.7k
4
votes
2
answers
1k
views
Sphere - Symmetry and Triangulation [closed]
The sphere is symmetric with respect to any rotation. However, it loses this property as soon as it is triangulated. Are there sequences of triangulations that possess particular large symmetry groups ...
- 7,276
0
votes
1
answer
73
views
Optimal radiating $(d{-}1)$-flats within a sphere
Permit me to revisit an earlier unresolved MO question,
"Chord arrangement that avoids confining small or large disks"
with a (very!) specific version, inspired by radiation therapy.
The main idea is ...
CommunityBot
- 1
14
votes
3
answers
966
views
Can a tangle of arcs interlock?
Can a (finite) collection of disjoint circle arcs in $\mathbb{R}^3$ be interlocked in the sense in that they cannot be separated, i.e. each moved arbitrarily far from one another while remaining ...
CommunityBot
- 1
7
votes
2
answers
907
views
Is there a 3d equivalent of this picture?
This question arises apropos of an earlier question I asked that was (VERY!!!) helpfully answered by Anton Petrunin:
Fitting a mesh to a density function
The picture below is the image of a regular ...
CommunityBot
- 1
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 ...
- 39.9k
2
votes
0
answers
586
views
Partitioning the Projective Plane
Throughout this post, by projective plane I mean the set of all lines through the origin in $\mathbb{R}^3$.
Side Note: If there are more standard definitions for any of the ideas presented here, ...
- 6,210
7
votes
1
answer
293
views
Can a tangle of arcs interlock in plane?
This is a variation of the question Can a tangle of arcs interlock?, asked by Joseph O'Rourke, and solved. I reproduce the question here:
Can a (finite) collection of disjoint circle arcs in $\...
CommunityBot
- 1
4
votes
1
answer
333
views
n-simplex in an intersection of n balls
Consider any $n$-simplex, $n \geq 2$. For each edge $(i,j)$, consider $n$-ball $B_{ij}$ such that vertices $x_i$ and $x_j$ are antipodal on this ball. Fix a point $x_0$ in the simplex. The question: ...
CommunityBot
- 1
2
votes
2
answers
284
views
Three questions concerning lattice points on sphere surfaces
Pardon my ignorance of this topic.
Q1.
In which dimensions $d$ is it the case that, for every natural number $n$,
there exists a sphere having exactly $n$ lattice points on it $(d{-}1)$-...
- 6,210
5
votes
1
answer
547
views
Cover of a n-simplex with balls
Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.
- 6,210
12
votes
7
answers
683
views
Can a tangle of arcs of ellipses interlock
This is a variation on an earlier question resolved by user35353: Can a tangle of arcs interlock? In that question, the arcs were restricted to circular arcs, and user35353's proof that one arc can be ...
CommunityBot
- 1
4
votes
2
answers
287
views
Problems similar to Borsuk’s Theorem in the plane
Consider a 2-dimensional Borsuk's theorem:
Every bounded set $S$ in the plane can be partitioned into three parts with diameter smaller than the diameter of $S$.
I wonder if there are any results ...
- 6,101
10
votes
2
answers
326
views
Do maximal polyhedra have algebraic volume?
Is it possible to prove that for every $n > 3$ the maximal possible volume of a convex polyhedron having $n$ vertices inscribed in a sphere of unit radius is an algebraic number?
Update: What can ...
- 6,819
5
votes
2
answers
563
views
Covering convex polygons with inscribed disks
The following problem came up when discussing mapping software (e.g., Google maps) with computer scientists. By $B(c,r)$ I mean the planar disk (open or closed, it doesn't matter) of radius $r$ around ...
- 151k
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
9
votes
1
answer
484
views
Which values can attain the minimum solid angle in a simplex
Given a simplex $S$ with a vertex $v$ by the solid angle at this vertex I mean the value $\hbox{vol}(B \cap S)/\hbox{vol}(B)$ where $B$ is a small enough ball centered at $v$ (for example, in the ...
- 976
13
votes
3
answers
835
views
What fraction of n-point sets in the unit ball have diameter smaller than 1?
This question is inspired by a recent talk by Matt Kahle on random geometric complexes.
Some simple notation: let $\mathcal{B} \subset \mathbb{R}^d$ be the unit ball in $d$-dimensional Euclidean ...
- 148k
4
votes
1
answer
184
views
What are interesting 3-colorings of the plane without rainbow lines?
This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, ...
- 12.1k
1
vote
1
answer
176
views
Helly's number from biconvex functions
Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \...
- 32.4k
2
votes
1
answer
171
views
Helly's Theorem for Biconvex Sets
Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
- 148k
16
votes
2
answers
1k
views
Integer lattice points on a hypersphere
Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice points ...
- 386
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 ...
- 1,083
2
votes
2
answers
255
views
What is the smallest number of subsets in such a subdivision?
Given any $30$ points in the plane, what is the smallest number of
subsets in a subdivision of the set of $30$ points into subsets such
that all the points in each subset are on the boundary of the ...
- 50.8k
0
votes
1
answer
229
views
Is this bounded?
May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a
convex polygon in the plane and $v_{m+1}$ be a vertex in the interior
of the convex polygon. Connect ...
- 1
4
votes
2
answers
567
views
Realization spaces for regular convex polytopes
Q1.
Are there convex polytopes combinatorially equivalent to each of the regular polytopes
that are realized with integer vertex coordinates?
&...
CommunityBot
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5
votes
4
answers
1k
views
Coloring Points in the Plane
Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed?
I heard this problem when I was a kid. Back then the most ...
- 14k
7
votes
3
answers
866
views
Not quite regular polyhedra
Take a naive interpretation of regular polyhedra:
All vertices (including epsilon ball) congruent
All edges congruent
All faces congruent
We can now find interesting families by removing one ...
- 3,532