**8**

votes

**1**answer

84 views

### A variant of Nelson-Hadwiger Problem on the chromatic number of the plane

The famous Nelson-Hadwiger problem asks about the chromatic number of the graph $G$, with the vertex set $V(G)={\mathbb R}^2$ where $a_1=(x_1,y_1), a_2=(x_2,y_2) \in V(G) \ $ form an edge iff ...

**3**

votes

**2**answers

607 views

### What's the name of this geometric mathematical modeling problem?

There is a right angle corner with width 1 in both directions. One wants to find the largest area shape which can pass through this corner.
I know that this is a famous problem, but what is it called?
...

**2**

votes

**0**answers

88 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 ...

**5**

votes

**1**answer

133 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 ...

**1**

vote

**1**answer

44 views

### number of affine pieces of linear interpolation of convex functions in high dimension

Consider a convex function $f$ defined on a $d$-dimensional hypercube $[0,1]^d$. Now for fixed $m \in \mathbb{N}$, consider the grids $\mathcal{G}_m=\{(i_1/m,\cdots,i_d/m)\}$ where ...

**3**

votes

**3**answers

335 views

### Aperiodic graphs

The concepts of being non-periodic and aperiodic for tilings have obvious versions for connected graphs with a countable set of vertices and a finite number of edges meeting at each vertex. A graph ...

**3**

votes

**1**answer

297 views

### Balancing real numbers in one dimension

Given numbers $0 \leq d_i \leq 1$ for $i=1,\ldots,m$, it is easy to see that you can always find signs $\varepsilon_i \in \{-1,1\}$ such that the partial sums $\sum_{i=1}^k \varepsilon_i d_i/2$, for ...

**3**

votes

**2**answers

148 views

### regular polyhedra (and polytopes) in hyperbolic geometry, and generalisations

While there exist regular tesselations of the hyperbolic plane with arbitrary regular polygons, there are no new regular polyhedra in hyperbolic (3D) space. This being quite trivial, it is probably ...

**9**

votes

**0**answers

234 views

### Neighborly family of coins

Here is a puzzle:
Find 5 identical coins. Can you arrange them so that every coin is touching every other coin?
The solution is here. The hint is: use the third dimension.
My questions are ...

**3**

votes

**0**answers

88 views

### Modeling bubble rafts

If you go to images.google.com and search on "bubble rafts", you'll see various pictures of disk packings that in large patches look like the six-around-one dense packing of the plane by equal-sized ...

**9**

votes

**1**answer

151 views

### Optimal shape for stabbing balls in $\mathbb{R}^3$

I have radius $r < \frac{1}{2}$ congruent balls with centers randomly distributed uniformly within a region,
say, within a unit-radius sphere $S$.
I shoot a ray/path through $S$, hoping to ...

**3**

votes

**0**answers

78 views

### Infinite counter-examples to Borsuk conjecture

When I formulate the Borsuk conjecture in the form of a coloring problem, Aigner told me that the formulation is not quite correct: It only works if the set is finite. He's right.
All the ...

**13**

votes

**1**answer

353 views

### Is the Ford disk packing optimal?

Given two unit-diameter disks tangent to a given line and to each other, determining a region bounded by two circular arcs and a line segment, is the Ford disk packing of that region the unique ...

**7**

votes

**0**answers

88 views

### Points on $k$ Circles

Let $k$ be a fixed positive integer. We want to find the minimum number $f(k)$, such that for a set of finite points in the plane, if any $f(k)$ of them are on $k$ circles, then all of them are on $k$ ...

**12**

votes

**0**answers

269 views

### Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...

**2**

votes

**0**answers

83 views

### degenerate abelian surfaces

I am wondering if the family of degenerate abelian surfaces constructed by K. Hulek, C. Kahn and S.H. Weintraub in "Moduli spaces of Abelian Surfaces: Compactification, Degenarations, and Theta ...

**3**

votes

**1**answer

163 views

### Packing bounds for sumsets, or, very discrete balls

Let $D\subset \mathbb{F}_2^n$ with $D=-D$ and $0\in D$. Write $k D$ for the set of all sums of $k$ (not necessarily distinct) elements of $D$. (This is the "ball" in the title.)
Now let $d(g,h)$ be ...

**1**

vote

**0**answers

77 views

### An exact fraction of a matrix

Let $A$ be a $n \times m$ real matrix with $n<<m$ and of rank $r<n$. It is known that $A$ has exactly two distinct non-zero singular values: $\sigma_{\max}$ and $\sigma_{2}$, and also that ...

**2**

votes

**1**answer

91 views

### Complex lines arrangements from a given wiring diagram

I'm working on some presentation of the fundamental group of the complement of a complex lines arrangement in $\mathbb{C}^{2}.$ In particular, in Arvola's article
"The fundamental group of the ...

**6**

votes

**2**answers

210 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 ...

**4**

votes

**0**answers

83 views

### Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries).
Can we partition this union into at most $n$ rectangles?
I think it's pretty ...

**3**

votes

**1**answer

142 views

### Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams.
I also found some relation with matroid theory. ...

**0**

votes

**0**answers

50 views

### Number of faces of polytope projecting to lower dimensional polyhedron

Denote $K=\mathrm{conv}(v_1, \ldots, v_n)\subsetneq\Bbb R^m$ to be convex set spanned by vectors $v_i\in\Bbb R^m$ with $m\leq n$ then what technique could be useful to upper bound minimum number of ...

**4**

votes

**2**answers

334 views

### How to flip one triangulation on a surface into another

Let $S$ be a compact orientable surface and $p_1,\dots, p_n\in S$ be distinct points. We consider all triangulations on $S$ with vertices $p_1,\dots, p_n$.
Is there an algorithm which takes two ...

**10**

votes

**1**answer

306 views

### Soft question: mathematics about truchet tiles

It seems that this is the first question on Truchet tiles on MO.
Shown above is a picture of a random tile, which you can see the resulting configuration is much like many membranes of cells.
I ...

**0**

votes

**0**answers

37 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 $D \subseteq \mathbb{R}^m$. ...

**1**

vote

**0**answers

155 views

### covering high dimensional hypercube by balls

suppose we are given the $d$-dimensional hypercube $H^d$ defined as
$$
H^d:=\left\{\sum_{i=1}^d\epsilon_ie_i:\ \epsilon_i\in \{0,1\}\mbox{ for }i=1,\dots , d\right\}
$$
and $(e_i)_{i=1}^d$ the ...

**8**

votes

**1**answer

115 views

### Tilting the $d$-cube to vertically separate its vertices

Let $C_d$ be a unit edge-length cube in $d$ dimensions.
I would like to orient it ("tilt" it) so that the vertical (last) coordinates
of its $2^d$ vertices are maximally separated, in the sense
that ...

**2**

votes

**1**answer

418 views

### Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or math.stackexchange.
You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be ...

**17**

votes

**1**answer

244 views

**14**

votes

**1**answer

332 views

### Are all well behaved “mean” functions on $\mathbb{R}^+$ equivalent?

Given a set $S$, a function $M: S\times S \rightarrow S$ is a mean if it satisfies the properties:
$M(a,a)=a\qquad$ (identity)
$M(a,b)=M(b,a)\qquad$ (commutativity).
and possibly
...

**2**

votes

**1**answer

91 views

### Measuring the Randomness and Statistics of Convex Polygons

How can I tell, how likely it is, that a given convex polygon with a sufficiently high number of edges is random and, if so, what kind of randomness it is (e.g. white noise)?
What is known about ...

**1**

vote

**2**answers

170 views

### Generalization of Bracketing (or one of its many equivalences)

I asked the following question on MathStackExchange, but I have not received any answers after almost 3 days. Although it may not be a research level question, I thought I could ask it here.
*"Is ...

**0**

votes

**0**answers

57 views

### Colorful version of Fisher's inequality for block designs

Is there such a thing? I am thinking of Karatheodory and Tverberg analogues here.

**3**

votes

**1**answer

89 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 ...

**1**

vote

**0**answers

52 views

### Can any Delone set be approximated by a model set?

Let $\Lambda \subset \mathbb{R}^d$ be a Delone set (uniformly discrete and relatively dense). I would like to know whether $\Lambda$ can be approximated by a model set in the Hausdorff distance.
...

**1**

vote

**0**answers

203 views

### Find m most distant points from a set of n points [closed]

I would like to find the $m$ (where $n$ $\geq$ $m$ > 1) maximally distant subset of points from a collection of $n$ $d$-dimensional points. Maximally distant means the sum of the pairwise distances ...

**6**

votes

**0**answers

111 views

### How big can a family of pairwise intesecting affine spaces be?

I apologize if this question might seem to be a bit too elementary.
Let $\mathbb{P}^n$ be the projective space over $k$ - an algebraically closed field of characteristic 0. Let $1\leq l\leq n-1$, and ...

**17**

votes

**0**answers

291 views

### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set ...

**4**

votes

**1**answer

206 views

### Panning for gold nuggets: a type of isoperimetric problem

Let $C$ be a unit-radius circle in the plane.
Suppose you have a total length $L$ of string available, and
your task is to connect chords of $C$ using no more
than $L$ of string to minimize the ...

**4**

votes

**1**answer

190 views

### The Universality Theorem by Mnev for uniform oriented matroids of rank 4 and higher

According to the Universality Theorem by Mnev (see below theorem 8.6.6 from [1]), for any open semialgebraic variety V there is a uniform oriented matroid of rank 3 whose realization space is stably ...

**3**

votes

**1**answer

130 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 ...

**8**

votes

**0**answers

161 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," ...

**6**

votes

**4**answers

216 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 ...

**6**

votes

**2**answers

237 views

### Counting valid coordinates

We are given a matrix $D = (d(i,j))_{1 \leq i,j \leq n}$ such that $d(x,z) \leq d(x,y) + d(y,z)$ for each $1 \leq x,y,z \leq n$. It is also known that $d(x,y) \in \mathbb{N}$ (In this question $0 \in ...

**6**

votes

**2**answers

337 views

### Are angles between points enough to decide the realizability?

Let n points in the plane be given whose coordinates we don't know.
Assume, however, that for any triple of the points we know the angle.
Question: Can we decide whether the n points are realizable ...

**7**

votes

**2**answers

397 views

### Visibility interpretation of Riemann zeta zeros on the critical line?

This is a long shot, but ...
The fraction of $\mathbb{Z}^2$ lattice points
visible from the origin
$1/\zeta(2)=6/\pi^2 \approx 61$%.
The fraction of $\mathbb{Z}^3$ lattice points visible
from the ...

**2**

votes

**1**answer

67 views

### Visibility kernels of embedded graphs

Let $G$ be a connected graph embedded in the plane with all edges straight segments.
For $\alpha \in (0,\pi)$, define an $\alpha$-path as a path in $G$ with
all turns at vertices within ...

**3**

votes

**2**answers

221 views

### A problem on chains of squares — can one find an easy combinatorial proof?

Consider the unit square $ S = [0,1] \times [0,1] $. For each $ n \in \mathbb{N} $, we can tessellate $ S $ by the collection
$$
A
= \left\{
\left[ \frac{i}{n},\frac{i + 1}{n} \right] \times
...

**3**

votes

**1**answer

123 views

### Points with pairwise integer distances in the plane

Consider $n>3$ points with pairwise integer distances in the plane! What is the relationship between these $n(n-1)/2$ integers? Do we have a theorem or result about these points? Does there exist a ...