**2**

votes

**0**answers

76 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

123 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 ...

**0**

votes

**0**answers

36 views

### Lattice isotopy type of uniform hyperplane arrangements

I am working on a problem related to the isotopy type of a certain class of hyperplane arrangements in $\mathbb{C}^{d}.$ For more references, compare Randell's work "Lattice-isotopic arrangements are ...

**9**

votes

**0**answers

220 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 ...

**1**

vote

**1**answer

22 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

317 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

**0**answers

84 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 ...

**3**

votes

**1**answer

280 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

107 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 ...

**4**

votes

**3**answers

256 views

### Perimeter/Neighborhood of a graph on grid

Hello,
I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one.
Now I want to claim ...

**3**

votes

**1**answer

235 views

### Min Bend Orthogonal Knots

I am seeking literature on 3D orthogonal drawings of knots,
especially minimum bend drawings.
An orthogonal drawing employs segments parallel to the axes of
a Cartesian coordinate system.
A bend is a ...

**13**

votes

**1**answer

339 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 ...

**9**

votes

**1**answer

143 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 ...

**2**

votes

**0**answers

70 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 ...

**7**

votes

**0**answers

81 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

246 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 ...

**1**

vote

**0**answers

66 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

132 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

74 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

75 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

163 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

79 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

130 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. ...

**10**

votes

**1**answer

280 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 ...

**3**

votes

**2**answers

302 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 ...

**7**

votes

**2**answers

680 views

### Which (semi)regular polyhedra are combinations of two others?

The convex combination of convex polytopes is a convex polytope.
An example in $\mathbb{R}^2$ is that a regular octagon
can be obtained as $\frac{1}{2} S + \frac{1}{2} S'$,
where $S$ is a square and ...

**3**

votes

**2**answers

80 views

### Average vertex degree in finite Delaunay triangulations in high dimensions

In $\mathbb{R}^2$ it's known that with a "random" point configuration, the average degree of a vertex in its Delaunay triangulation is 6.
Does anyone know of a similar result in higher dimension? I ...

**0**

votes

**0**answers

43 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 ...

**16**

votes

**1**answer

451 views

### Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$,
with coordinates
$\equiv 2 \bmod 3$,
we place, with equal probability, one of these six patterns:
The result ...

**3**

votes

**1**answer

164 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 ...

**0**

votes

**0**answers

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

**2**

votes

**1**answer

89 views

### mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...

**1**

vote

**0**answers

142 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

111 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

216 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

235 views

**11**

votes

**1**answer

285 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
...

**4**

votes

**1**answer

212 views

### convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where ...

**36**

votes

**3**answers

2k views

### Is the “Napkin conjecture” open? (origami)

The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...

**13**

votes

**2**answers

979 views

### Are the Platonic solids shadows of 4-polytopes?

Say that a 3D shadow of a 4-polytope is a parallel projection to 3-space, not necessarily orthogonal to that 3-space (that would make it an orthogonal projection).
I am wondering if each of the five ...

**2**

votes

**1**answer

75 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

166 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

55 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

79 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 ...

**7**

votes

**0**answers

180 views

### Minimal “basis” in $n$ dimensional unit cube

Let's
$$
B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots
$$
and let's
$$
C\subseteq B^n,
$$
$$
S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...

**1**

vote

**0**answers

50 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.
...

**25**

votes

**5**answers

2k views

### Nonconvex manhole covers

One common reason given for the circularity of manhole covers is that they can't fall through the manhole. For convex manhole covers, this property is equivalent to having constant width — if ...

**1**

vote

**0**answers

161 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 ...

**17**

votes

**0**answers

267 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 ...

**6**

votes

**0**answers

107 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 ...