**32**

votes

**0**answers

483 views

### Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...

**15**

votes

**0**answers

1k views

### Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...

**14**

votes

**0**answers

280 views

### Knight's tours in higher dimensions

I wonder if Knight's Tours have been explored in higher dimensions,
using the following definition of a knight move.
In dimension $d=2$, the knight moves left/right and forward/back
one step and two ...

**11**

votes

**0**answers

349 views

### Drawings of complete graphs with $Z(n)$ crossings

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor ...

**11**

votes

**0**answers

415 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**10**

votes

**0**answers

274 views

### Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems
on convex polyhedra.
Progress has been made on several of his problems, and perhaps some have been completely ...

**9**

votes

**0**answers

291 views

### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

**8**

votes

**0**answers

117 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**8**

votes

**0**answers

117 views

### Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...

**7**

votes

**0**answers

75 views

### Is the equidissection spectrum closed under addition?

If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area?
(I'm editing after realizing that my conjecture that a ...

**7**

votes

**0**answers

333 views

### Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this ...

**7**

votes

**0**answers

140 views

### Fractional Helly for more than one piercing

Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...

**7**

votes

**0**answers

227 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 ...

**7**

votes

**0**answers

268 views

### 3-dimensional Cayley graph

I would like to see Cayley graphs drawn in 3-dimensional Euclidean space such that the vertices are represented by points and various shadows display the actions of the generators.
For example, ...

**6**

votes

**0**answers

165 views

### Linked circles in R3

Two circles in 3-D are linked iff each one passes through the interior of the other.
There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique ...

**6**

votes

**0**answers

789 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 ...

**6**

votes

**0**answers

260 views

### A question about a blue fan and a red fan and their common refinement

Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of ...

**6**

votes

**0**answers

211 views

### balls in arrangements of hyperplanes

The following theorem is from Aronov, Naiman, Pach and Sharir's
An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem ...

**5**

votes

**0**answers

402 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 ...

**5**

votes

**0**answers

428 views

### Optimal Gear Trains

Suppose you need to slow down a turning motor so that a gear turns at
an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and
$b$ are natural numbers. For example, this set of ...

**4**

votes

**0**answers

224 views

### Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset S of the xy-plane is star-convex if there is ...

**4**

votes

**0**answers

119 views

### Graph drawing maximizing the volume of the convex hull

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that ...

**3**

votes

**0**answers

55 views

### A taut string of equilateral triangles

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each
of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.)
Think of $T$ as a physical, rigid ...

**3**

votes

**0**answers

89 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 $\rho$ ...

**3**

votes

**0**answers

121 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 ...

**3**

votes

**0**answers

121 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 ...

**3**

votes

**0**answers

236 views

### Coloring $\mathbb{Z}^k$ and a fixed point theorem

This is potentially another approach to this question. I put it as an update there, but perhaps it would be better to post it separately. If we color $\mathbb{Z}^k$ with the $\ell_\infty$ metric in ...

**3**

votes

**0**answers

131 views

### A fast way to test whether a partial function can be extended to a chirotope of rank 3 ?

Hello,
What is a fast way to test whether a partial function can be extended to a chirotope of rank 3 ? That is: I have a domain
$E = \{1,...,n\}$
and a partial function
$f: E^3 \to \{-1, 0, 1\}$
...

**3**

votes

**0**answers

125 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 ...

**2**

votes

**0**answers

148 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 ...

**2**

votes

**0**answers

43 views

### Looking for N-dimensional spheres in the configuration space of the colorful Tverberg problem

Here we use standard notation for Tverberg's theorem: Dimension $d$, number of partition blocks $r$, and $N=(r-1)(d+1)$.
The configuration space of Tverberg's theorem is the simplicial complex ...

**2**

votes

**0**answers

326 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, ...

**2**

votes

**0**answers

77 views

### Criteria to decide whether a subset of the boundary complex of a polyhedron is a manifold?

Let $\mathcal{P} \subseteq \mathbb{R}^d$ be a convex polyhedron. Let $K$ be a subset of the boundary complex of $\mathcal{P}$. (Perhaps $K$ could be defined in terms of a system of linear ...

**2**

votes

**0**answers

126 views

### A question on discrete numerical simulation on fluids mechanics

I read the paper "Stable, circulation-preseving simplicial fuids" by Elcott, et al: http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretization of fluids. I have ...

**2**

votes

**0**answers

88 views

### Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime,
on neither the $x$- nor the $y$-axis.
So $a^2+b^2$ is a prime.
Construct a region $D(z)$ surrounding $z$ which is the
largest ...

**2**

votes

**0**answers

96 views

### Rational viewing points in a polygon

We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside ...

**2**

votes

**0**answers

236 views

### Erdős-Szekeres empty pseudoconvex $k$-gons

I am wondering if the
Erdős-Szekeres
empty convex $k$-gon question has a different answer if
convexity is replaced by a pseudoline-version of convexity.
The empty convex $k$-gon question
is a variant ...

**2**

votes

**0**answers

138 views

### Minimally 6-connected 3D discrete lines that are convex lattice sets

There are several definitions of 3D discrete lines, e.g. http://diwww.epfl.ch/w3lsp/publications/discretegeo/nratddl.html , http://dx.doi.org/10.1007/978-3-642-19867-0_4 . However, I know of none that ...

**2**

votes

**0**answers

185 views

### Is there any symmetric unshellable triangulation of a tetrahedron?

M. Rudin gave a triangulation of a tetrahedron with the property that after any small tetrahedron removed, the remaining part is not homeomorphic to a ball (An unshellable triangulation of a ...

**1**

vote

**0**answers

31 views

### Moment lemma for circular arc polygons

A "moment lemma" originally due to Fejes Tóth (I guess) states that if $P$ is any polygon (in $\mathbb{R}^2$) with $k$ sides, then for any $\mathbf{y}$, we have ...

**1**

vote

**0**answers

51 views

### Approximating Unit covering of d-dimensional points

Given a $d$-dimensional disk of radius $2$ in $\mathbb{R}^d$, how many disks of radius $1$ suffice to cover it. Of course, it's fine if the smaller disks overlap. What matters is to specify a finite ...

**1**

vote

**0**answers

45 views

### Euclidean embedding of a graph based on 1-ring neighborhood distances only

Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...

**1**

vote

**0**answers

52 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 ...

**1**

vote

**0**answers

74 views

### Group actions on polytopes in indefinite integer lattices

Is anything at all known about polytopes in indefinite integer lattices? I'm interested in lattice automorphisms which preserve certain polytopes of "high regularity" (e.g. cones). As a first step, ...

**1**

vote

**0**answers

115 views

### Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular:
Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...

**1**

vote

**0**answers

157 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, ...

**1**

vote

**0**answers

146 views

### When is the conical hull of a finite set of vectors a subset of the space? (and tilings)

Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...

**0**

votes

**0**answers

114 views

### “Open Points” in the 1983 proof of Szemerédi-Trotter theorem

I was reading through the 1983 paper "Extremal Problems in Discrete Geometry" and I was confused about the definition of "open point" appearing in this paper. By this point in the paper, the authors ...

**0**

votes

**0**answers

48 views

### Paths on Cartesian products of graphs satisfying linear constraints

Assume integers $d > r > 0$ and a connected graph $G$ with $d$ vertices. Every point on the $r$-fold Cartesian product of $G$ with itself, $G^{\square r}$, is equivalent to a dimension-$d$ ...

**0**

votes

**0**answers

37 views

### Covering the annulus of symmetric convex body

Consider a symmetric convex body $A$ in $R^d$. Now, we draw another object, $A'$, concentric and translated with respect to A and having radius slightly greater than twice to the radius of $A$.
Now ...