All Questions
Tagged with reference-request discrete-geometry
47 questions with no upvoted or accepted answers
24
votes
0
answers
760
views
How much of the plane is 4-colorable?
In 1981, Falconer proved that the measurable chromatic number of the plane is at least 5. That is, there are no measurable sets $A_1,A_2,A_3,A_4\subseteq\mathbb{R}^2$, each avoiding unit distances, ...
- 7,633
21
votes
0
answers
453
views
Does every 5-celled animal tile the plane?
An animal in the plane is a finite set of grid-aligned unit squares in $\mathbb{R}^2$. (The definition is the same as a polyomino, but where we relax the connectivity requirement.) One may ...
- 3,068
15
votes
0
answers
477
views
Expanding disks lead to what packing of the plane?
Suppose one sprinkles points uniformly at random on the infinite Euclidean plane,
with some density $\rho$ per unit area.
View the points as disks of radius zero.
Now the radii $r$ of all disks grows ...
- 151k
10
votes
0
answers
1k
views
Interpolating points with minimum curvature constraint
I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...
- 151k
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
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
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
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
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
0
answers
234
views
To whom is the classification of atomic, modular finite lattices due?
Here lattice means a poset with meets and joins. A lattice is called atomic if every element is a join of atoms. There are a few different ways to define modular for finite lattices: one is that the ...
- 24.2k
4
votes
0
answers
60
views
How are these "Voronoi-dual" configurations called?
If $\mathscr P\subset \mathbb R^d$ is a discrete point configuration, take the Voronoi diagram of $\mathscr P$ and call $\mathscr P'$ the vertices of that diagram.
I would like to know if ...
- 2,041
4
votes
0
answers
230
views
Is this case of Barnette's Conjecture known?
Context: Barnette's Conjecture is that every bipartite cubic polyhedral graph is Hamiltonian. I have been interested by this problem for a long time, and I recently came up with a result. From my ...
- 3,499
4
votes
0
answers
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
213
views
Counting the polytopes of the translates of the resonance hyperplane arrangement inside the unit hypercube
Let $n$ be a positive natural number. For all $\emptyset \subset S \subseteq \{1, \ldots, n\}$ and $k \in \mathbb{Z}$, define the hyperplane $H(S,k)$ in $\mathbb{R}^n$ given by the equations
$$H(S,k):=...
- 283
4
votes
0
answers
164
views
Two variants of the Littlewood-Offord theorem
I found two different looking things being called the Littlewood-Offord theorem,
If $\vec{a} \in \mathbb{R}^k \setminus 0$ and $t \in \mathbb{R}$ then there are $O(\frac{2^k}{\sqrt{k}})$ points $x \...
- 2,246
4
votes
0
answers
143
views
Balanced partitions of vector sets
We are interested in the following
Lemma. Let $V\subset [0,1]^n\subset\mathbb R^n$ be a set of $n$-dimensional vectors. Then for each $r\le |V|$ there exists a partition $$V=V_1\cup V_2\cup\dots \cup ...
- 5,409
4
votes
0
answers
158
views
Reference for the notion of polyhedra "degenerations"
Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ (...
- 3,513
4
votes
0
answers
189
views
Slices of Simplices that are Simplices, Reference?
I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...
- 41
3
votes
0
answers
208
views
Reference request: Carathéodory-type theorem for convex hulls of closed sets
I'm looking for a reference for the following theorem.
Theorem Let $X$ be a closed subset of $\mathbb{R}^N$, and let $a$ be a point of its convex hull $\operatorname{conv}(X)$. Then there exist ...
- 27.7k
3
votes
0
answers
93
views
Minkowski problem for polytopes: the origin of necessary condition
Minkowski's uniqueness theorem for polytopes concerns the specification of the shape of a polytope by the directions and measures of its facets.
Theorem (Minkowski). Let $A_i$ be positive faces areas ...
- 12.3k
3
votes
0
answers
116
views
A theory of refined h- and f-polynomials for the permutahedra, associahedra, noncrossing partitions, and tropical Grassmannians (references)
Looking for references (insights) on a theory encompassing a notion of refined face polynomials and their associated refined h-polynomials that are generalizations of the relation between ordinary f-...
- 10.5k
3
votes
0
answers
86
views
Sums over lattice points in homogeneously expanding domains
In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...
- 3,107
3
votes
0
answers
135
views
Intersecting the unit n-cube and (n-1)-planes
(Is this a known problem?)
Question Let $\ 1<n\in\mathbb N.\ $ What is the greatest $(n-1)$-area
$\ S(n)\ $ of $\ L\cap I^n\ $ where $\ I^n\subseteq\mathbb R^n\ $ is the unit cube, and $\ L\ $ ...
- 4,786
3
votes
0
answers
144
views
Counting homologically non-trivial and trivial cycles in $n \times n$ square lattice torus of a given length $l \geq n$
This should be a fairly standard question but I can't really seem to find a reference.
Consider an $n \times n$ square lattice torus $\mathbb T$. Given a length $l \geq n$, what is the number of ...
- 269
3
votes
0
answers
169
views
Computing Voronoi poles in $\mathbb{R}^d$ (the farthest points within each cell)
Say I have a Voronoi diagram of some points $p_1,\dots,p_n\in\mathbb{R}^d$, which tesselates $\mathbb{R}^d$ into cells $V_1,\dots,V_n$. Within each cell $V_i$, the pole is defined as the vertex of $...
- 41
3
votes
0
answers
391
views
Dissection of a polygon into convex polygons
Problem: for a fixed integer $m\geqslant 3$ find all $n$ such that no $n$-gon can be dissected into convex $m$-gons.
I would be very grateful for any information on this problem.
Remark 1. There ...
- 935
3
votes
0
answers
142
views
Dimension of convex arrangements for hypergraphs
Suppose you have a hypergraph H on n vertices. Let d be the smallest integer such that we can find an arrangement A of convex subsets in Rd so that H represent the intersections of sets in A.
Has ...
- 4,586
2
votes
0
answers
65
views
Structure Theory for Tree Decompositions
I that $G=(V,E,W)$ is a weighted graph with positive edge weights and a finite set of vertices $K$. Let $0\le k,M\le K$ be a fixed integer.
Is is known when $G$ admits the following type of ...
- 21
2
votes
0
answers
77
views
Flexagons and noncrossing partitions
Turns out a couple of series related to the faces of flexagons
popped up in my explorations of combinatorial reciprocities in a group algebra for sets of partition polynomial (ParPs) related to the ...
- 10.5k
2
votes
0
answers
233
views
Do you know this formula for the scalar product in barycentric coordinates?
I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it?
Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
- 1,048
2
votes
0
answers
47
views
Source request: Optimal bounds on signings of points from a convex body
I recently came across an old survey of problems in discrete geometry: https://pdfs.semanticscholar.org/c350/f4d4a9466fa6708d99ec1187c63d89bed20f.pdf
Problem 2.1 from the list caught my eye. It states ...
2
votes
0
answers
87
views
A theory of (or reference for) symmetric point arrangements
I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
- 13.6k
1
vote
0
answers
76
views
Shellable non-pseudomanifolds with dimension greater than 2
Shellability of simplicial balls and spheres (simplicial complexes whose geometric realizations are homeomorphic to balls and spheres) has been studied quite extensively. There are many explicit ...
1
vote
0
answers
145
views
Lower bound $|\sum_{x \in X} \phi(x) - \int_{\mathbb{R^2}} \phi(x) \, dx | \geq C f(\phi)$
I asked this question on math.stackexchange before, but with a bad formulation. I think the problem is quite complicated, so I decided to ask it here. Tell me if I shouldn't.
Very recently, I ...
- 183
1
vote
0
answers
69
views
Can sufficiently symmetric polytopes be uniquely reconstructed from their 1-skeleton?
General convex polytopes can not be uniquely reconstructed from their 1-skeleton1, as explained here. Not even the dimension is known from the skeleton, as e.g. the complete graph $K_n,n\ge 5$ is the ...
- 13.6k
1
vote
0
answers
278
views
Sphere packings with antipodal (unequal) spheres
Let $\|\cdot\|_2$ denote the Euclidean norm, let $\langle \cdot, \cdot\rangle$ denote the standard dot product, and let $\mathcal{S}^{d-1} = \{\mathbf{x} \in \mathbb{R}^d: \|\mathbf{x}\|_2 = 1\}$ ...
- 733
1
vote
0
answers
71
views
Name for a Specific Planar Linear Transformation
Is there a name for linear transformations of the plane, that make $4$ points in general convex configuration co-circular, with the biggest circle through those points and, how can they be determined ...
- 13.2k
1
vote
0
answers
39
views
Homology of the subcomplexes of the "diamond shaped" sphere under 1-norm in $R^n$ as a simplicial complex
The 1-norm on $\mathbb{R}^n$ is defined by $\|v\| = |v_1| + |v_2| + \cdots + |v_n|$ for a vector $v = (v_1, \ldots, v_n) \in \mathbb R^n$.
The unit sphere $S^{n-1}_1$ under the 1-norm is a simplicial ...
- 459
1
vote
0
answers
77
views
Non-adjacent Pair of Edges with Minimal Weight Sum
Given an weighted, undirected Graph $G(V,E)$ without loops or parallel edges,
what is the complexity of determining a pair of non-adjacent edges, whose sum of weights is w.l.o.g. minimal?
is that ...
- 13.2k
1
vote
0
answers
70
views
Covering number of the range of a function
I have come across the need to know a bound on a certain curious quantity: the covering number of the range of a continuous function $f: D \rightarrow \mathbb{R}^n$, where $D \subseteq \mathbb{R}^m$. ...
- 183
1
vote
0
answers
61
views
Generalizing Concepts of Planar Euclidean Geometry to Symmetric TSP-Instances
To me it seems possible, to successfully look at symmetric TSP instances from a geometry-point of view.
Examples are:
the diagonals of the convex hull of a set of points in the euclidean plane; ...
- 13.2k
1
vote
0
answers
193
views
Lattice-point enumeration question involving linear combinations of matrices
I would like to know some references to learn more about an answer to this question, if there are any references:
Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, \...
- 170
1
vote
0
answers
371
views
Simple development of simple curve on a cone
Let $\Lambda$ be a cone with apex $a$ and apex angle $\alpha$. Draw a simple (non-self-intersecting)
curve $C=(x,y)$ on $\Lambda$, and then develop it to a curve
$\overline{C}$ on a plane by rolling $...
- 151k