All Questions
Tagged with discrete-geometry reference-request
174 questions
9
votes
4
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474
views
Minimum number of common edges of triangulations
Let $S$ and $T$ be two triangulations.
We define
$c(S,T)$ as the number of edges shared by $S$ and $T$.
With this, we can define
$f(n) = \min_{P} \min_{S,T} c(S,T)$.
Here the first minimum goes over ...
- 479
9
votes
3
answers
1k
views
Generalization of Sylvester-Gallai theorem
The Sylvester-Gallai theorem states that it is not possible to arrange a finite number
of points so that a line through every two of them passes through a
third unless they are all on a single ...
9
votes
2
answers
505
views
Moore graphs and finite projective geometry
In a comment on a blog post from 2009 about the hypothetical Moore graph(s) of degree 57 and girth 5, Gordon Royle offered the following observation (reproduced here in full for the sake of ...
- 1,645
9
votes
1
answer
460
views
Connections between linear representations and permutation representations
A finite group $\Gamma$ might be represented by a linear transformation
$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$
or by permutations
$$\phi :\Gamma\to\mathrm{Sym}(n).$$
Of course, latter ones can ...
- 13.6k
9
votes
0
answers
100
views
A characterization of root systems via their intersections with halfspaces
In a recent preprint I obtained a nice characterization of root systems as a side product.
I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
- 13.6k
8
votes
4
answers
530
views
Inside-out polygonal dissections
A dissection of a polygon $P$
is a partition of $P$ into a finite number of pieces, which can then be rearranged
(via planar translations and rotations) and joined (without overlap) to form a new ...
- 151k
8
votes
1
answer
885
views
Maximal tetrahedra inscribed in ellipsoid
Pietro Majer quoted the theorem of Michel Chasles in his MO question,
"Convex curves with many inscribed triangles maximizing perimeter,"
which states that the triangles of maximum perimeter inscribed ...
- 151k
8
votes
0
answers
358
views
Coloring toroidal polyhedra with convex faces?
Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 non-...
7
votes
1
answer
938
views
Which knots' stick numbers are twice their crossing numbers?
Looking at a table of minimum stick numbers for knots (table here),
it seems the known upper bound of $2 c(K)$ in terms of the knot crossing number $c(K)$
is realized by the trefoil $3_1$—it ...
- 151k
7
votes
3
answers
551
views
Minkowski's theorem for non-0-symmetric sets
Let $\Lambda \subseteq \mathbb{R}^n$ be a full-rank lattice, i.e. $\Lambda = A \mathbb{Z}^n$ for some $A \in \mathrm{GL}_n (\mathbb{R})$, and let $C \subseteq \mathbb{R}^n$ be a $0$-symmetric convex ...
user92170
7
votes
4
answers
377
views
Discretizing a line segment with pixels which satisfies the Pythagorean theorem
There are plenty of line drawing algorithms to discretize line segments using pixels.
The Bresenham's algorithm gives a line where the number of pixels in the segment is the same as its width (in x-...
- 15.8k
7
votes
3
answers
805
views
Wrapping a convex polyhedron with string
This is a meta-question, rather than a specific mathematical question.
I am seeking a mathematical definition that captures the following physical idea.
Suppose you have a convex polyhedron $P \...
- 151k
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
1
answer
1k
views
Elementary precise estimate of the covering number of euclidean balls by hypercubes
I am looking for a straightforward way to upper bound the covering number of a $d$-dimensional euclidean ball by $\ell_\infty$-balls of radius $\varepsilon$, which I will call cubes of sidelength $2\...
- 172
7
votes
1
answer
299
views
Lipschitz-continuity of convex polytopes under the Hausdorff metric
Recently, I proved the following Lipschitz-continuity like result for convex polytopes:
Let $A\in\mathbb R^{m\times n}$ and $b,b'\in\mathbb R^m$ be given such that $\{x\,:\,Ax\leq 0\}=\{0\}$ (which ...
7
votes
1
answer
216
views
How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?
This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...
- 1,350
7
votes
0
answers
187
views
distance distributions on a hypersphere?
Fix a real number $0\leq t\leq 1$ and an integer $n>1$. Let
$\mathbb{S}^{n-1}\subset\mathbb{R}^n$ denote the unit hypersphere. Define
$$d_N(n;t):=\max\sum_{i<j}\Vert P_i-P_j\Vert_2^t$$
where ...
- 43.2k
6
votes
2
answers
546
views
On circles and ellipses drawn on an infinite planar square lattice
Consider a plane with a square lattice formed by all points with both coordinates as integers. As can be easily seen, a simple parabola can be found that passes through infinitely many of the square ...
- 5,979
6
votes
4
answers
2k
views
Delaunay triangulations and convex hulls
This is a reference request.
I have the impression that those who work in computational geometry are accustomed to the following. You have some locally finite set of sites in $\mathbb{R}^n$ and you ...
6
votes
2
answers
364
views
Triangles whose vertices and center have all the same color
A plane is colored with two colors. It's an easy exercise to prove that it's always possible to find an equilateral triangle whose vertices have all the same color.
Does anyone know any proof or ...
- 3,153
6
votes
2
answers
400
views
Geometric dominating set: NP-complete?
Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are
the Euclidean distance between its endpoint vertices.
Say that a set of vertices $D \subseteq V$ is a geometric ...
- 151k
6
votes
2
answers
381
views
Lattice-cube minimal blocking sets
Let $C_d(n)$ be the lattice cube consisting of the $n^d$ points with
each of its $d$ coorindates in $\lbrace 1,2,\ldots,n \rbrace$.
Define a blocking set for a lattice cube to be a set of points
in ...
- 151k
6
votes
2
answers
424
views
A class of tilings with amazing visual qualities
For more examples please see my related question on MSE:
Interesting tiling with a lot of symmetrical shapes
This is achieved by rotation of square grid over itself by atan(3/4).
Resulting ...
- 161
6
votes
1
answer
587
views
Study of convex polytopes via commutative algebra
Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...
- 211
6
votes
1
answer
276
views
Matching on sphere to create cycle with chords
Imagine a number of chords of a sphere $S$ which nearly, but not quite, pass through
the center of $S$, in such a way that no pair of chords intersect:
I would like to ...
- 151k
6
votes
1
answer
666
views
Pick's Theorem for rational points of bounded height
I wonder if the various lattice-point theorems, such as
Pick's Theorem or
Minkowski's Lattice Theorem,
have been generalized to the collection of points
with rational coordinates no more than height ...
- 151k
6
votes
0
answers
132
views
Have the affine simplicial line arrangments been enumerated?
I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements.
A line arrangment is a family of straight lines in $\Bbb R^2$. It is simplicial if all regions are ...
- 13.6k
6
votes
0
answers
118
views
Convex hull of all-ones principal submatrices
For a subset $S$ of $\{1,\ldots,n\}$,
let $\mathbf{1}_S\in\{0,1\}^n$ denote the indicator vector of $S$, with a $1$ on the $i$th coordinate iff $i\in S$. Let $\mathcal{X}$ denote the convex-hull of ...
- 617
6
votes
1
answer
295
views
A conjecture (or theorem?) on unit vectors in a Euclidean space
I have heard (if I am not mistaken) that there exists the following conjecture (or theorem?).
Let $u_1,\dots,u_n$ be unit vectors in an $n$-dimensional Euclidean vector space. Then there exists ...
- 21.8k
5
votes
2
answers
373
views
Genus of Tutte-Coxeter Graph
What is the genus of the Tutte-Coxeter graph -- the incidence graph of the
GQ of order 2? Seems like it should be well known, since nearly every other
parameter for that graph is known, but I can ...
user48028
5
votes
1
answer
266
views
Contracting a set to a ball
$\newcommand\R{\mathbb R}\newcommand\S{\mathbb S}$
Question 1: Let $S$ be a nonempty measurable subset of $\R^n$. Let $B$ be a closed ball in $\R^n$ such that $m(B)=m(S)$, where $m$ is the Lebesgue ...
- 128k
5
votes
2
answers
307
views
Tiling a Jordan polygon
I saw this problem some years ago, don't remember the source:
Let $P$ be a Jordan polygon (i.e. the only points of the plane belonging to two edges are the polygon vertices) that can be tiled with ...
- 3,153
5
votes
2
answers
342
views
Minimum length of a convex lattice polygon containing k lattice points?
Let $f(k)$ denote the minimum length of a convex lattice polygon containing exactly $k$ lattice points (including lattice points on the boundary).
It is not too hard to show that $k = \frac{1}{4\pi} ...
- 191
5
votes
1
answer
398
views
Ham sandwich theorem for discrete measures - reference request
A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):
For every $\mu_1,...,\mu_n$ discrete (i.e., purely atomic) ...
- 1,163
5
votes
2
answers
441
views
Touching-tetrahedra graphs
Have the graphs representable by touching tetrahedra been explored?
Let $\cal T$ be a collection of tetrahedra in $\mathbb{R}^3$
with pairwise disjoint interiors.
Define a graph $G_{\cal T}$ to have ...
- 151k
5
votes
2
answers
333
views
Characterization of combinatorial manifolds in terms of links
I need to reference the following result. Do you know a good source?
The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent:
a) $S$ is an $n$ manifold;
b) The link of ...
- 5,019
5
votes
1
answer
245
views
What is Known About the Complexity of Calculating Minimal Surface Polyhedra?
I am currently ruminating about ways of generalizing Minimum Spanning Trees to Minimum Spanning "Hypertrees", where the cost is associated with simplex volumes and, where certain topological ...
- 13.2k
5
votes
1
answer
190
views
Finding a superbase in a lattice of Voronoi first kind
An $n$-dimensional lattice in $\mathbb R^n$ is said to be of Voronoi’s first kind if it there exists $n+1$ vectors $b_1,\cdots b_{n+1}$ (called the superbase) such that
$\{b_1,\ldots,b_n \}$ is a ...
user14875
5
votes
1
answer
161
views
Upperbounding the number of regions induced by a set of unit disks
Given a set $D$ of $n$ same radius disks, embedded in the plane, their arrangement induces a number $k$ of connected regions in $\mathbb{R}^2 \setminus \cup_{d \in D}$ .
I am interested in an upper ...
- 53
5
votes
1
answer
265
views
Maximal number of triple intersection points of $n$ circles
It is easy to show that $n$ (mutually different) circles on the plane can have maximum $n(n-1)$ intersection points. In our optimal graph drawing research we have encountered a counterpart of this ...
- 5,409
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 ...
- 151
4
votes
2
answers
349
views
How many dihedral angles need to be specified to uniquely specify a triangulated polyhedron?
Suppose you are given a simplicial complex $K$ homeomorphic to the sphere and for each each edge of the complex a label specifying a length of that edge (this gives us a polyhedral metric on $K$). In ...
- 185
4
votes
2
answers
482
views
Generators of a 2D lattice
Dear MO_World,
I'm hoping someone can point me towards a reference for something. I have an invertible $2\times 2$ matrix, $A$, with real entries such that for both of the rows, the entries are ...
- 23.2k
4
votes
1
answer
493
views
Counting number of points on a lattice in a hypercube
Suppose I have a lattice $\Lambda \in \mathbb{R}^n$. Let $X_i >0$ for $i=1,..,n$. I am interested in some references regarding counting number of points of $\Lambda$ inside $[-X_1, X_1] \times \...
- 3,625
4
votes
2
answers
425
views
Algorithm for Reconstructing Point Sites from a Voronoi Diagram
how can one construct a finite set of points in the euclidean plane from its Voronoi Diagram and, what is the complexity of the problem?
- 13.2k
4
votes
2
answers
1k
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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 ...
- 1,256
4
votes
2
answers
173
views
4-polytopes with only one kind of regular facet
Is there a neat way to show (or a reference that already proves) that
the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes?
the 24-cell is the only convex 4-polytope in ...
- 13.6k
4
votes
1
answer
365
views
Orchard-planting problem in space
The original orchard-planting problem asks for the maximum number of $3$-point lines attainable by a configuration of points in the plane. I am interested in its natural generalization for (three-...
- 5,409
4
votes
1
answer
323
views
What properties does generalized Delaunay triangulation have?
Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
- 19k
4
votes
1
answer
646
views
Combinatorial geodesics
[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.]
I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics &...
- 9,720