**8**

votes

**0**answers

101 views

**10**

votes

**1**answer

251 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

68 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

154 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

52 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

74 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

45 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

126 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

171 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 ...

**6**

votes

**0**answers

102 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 ...

**-2**

votes

**0**answers

76 views

### Minimum distance in unit cube

Given the unit cube $I^n=[0,1]^n$, let $x_1,\cdots,x_n\neq (0,0,\cdots,0)$ be $n$ linearly independent vertices (corners) of $I^n$, and $p(x_1, \cdots, x_n)$ be the projection of $(0,0,\cdots,0)$ onto ...

**17**

votes

**0**answers

245 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

193 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

128 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

115 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

118 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

181 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

227 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

315 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

377 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

64 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

205 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

86 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 ...

**6**

votes

**1**answer

218 views

### Thinnest 2-fold coverings of the plane by congruent convex shapes

It is an unsolved problem to determine the "thinnest" $2$-fold covering of
the plane by disks.
The $2$-fold coverage problem by disks is to find the minimum number of congruent
(unit-radius) disks ...

**3**

votes

**1**answer

49 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 ...

**8**

votes

**1**answer

327 views

### Coloring of the plane

I would like to know the minimum number k such that the plane R^2 can be coloured with k colors such that no colour contain all the possible distances. In other words, a colouring such that each color ...

**3**

votes

**0**answers

62 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 ...

**2**

votes

**0**answers

152 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 ...

**4**

votes

**2**answers

109 views

### (non-)existence of the aperiodic monotile

The aperiodic monotile problem asks whether there exists a single tile that every tiling made with it results non-periodic. What is known about this problem? If this tile exists, how can it be/not be? ...

**7**

votes

**1**answer

137 views

### How many maximal triangulations of a rectangle?

I posted the following question on MathStackExchange, but I didn't any answer. So please let me post it on MathOverflow.
Let $L_{m,n}\subset\mathbb R^2$ be a rectangle given by $[0,m]×[0,n]$ with ...

**28**

votes

**2**answers

563 views

### what-if.xkcd.com: stabbing (simply connected) regions on the 2-sphere with few geodesics

In the latest what-if Randall Munroe ask for the smallest number of geodesics that intersect all regions of a map. The following shows that five paths of satellites suffice to cover the 50 states of ...

**6**

votes

**2**answers

117 views

### Number of edges in linklessly embeddable graphs

Consider graphs over $n$ nodes.
What is the maximum number of edges of a linklessly embeddable graph?
A more general question is the following. Given $\mu$ what is the maximum number of edges of ...

**22**

votes

**2**answers

575 views

### Random points on the unit sphere

Suppose you have $n$ points picked uniformly at random on the surface of $\mathbb{S}^d,$ and let the volume of the convex hull of these points be $V_{n, d}.$ Clearly, $V_{n, d}$ converges to the ...

**6**

votes

**2**answers

181 views

### Pictures of the von Neumann polytope

Are there any graphic portrayals of von Neumann polytopes in low dimensions?

**6**

votes

**1**answer

333 views

### Groups and pregeometries

Definition.
For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say
that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous ...

**2**

votes

**1**answer

105 views

### Distance from constant width bodies

EDIT As @David has observed, my conjecture was clearly wrong for $\ n:=2.\ $ Let me still give it a chance for $\ n\ge 3$.
I'll call a family $\ F\ $ of bound closed convex subsets of $\ \mathbb ...

**6**

votes

**1**answer

147 views

### Minimizing deep holes in sphere packings

What's the current state of knowledge regarding packings of spheres in $n$-space that minimize the supremum of the sizes of the holes? This notion of tightness is more rigid than asymptotic density. I ...

**5**

votes

**1**answer

119 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) ...

**3**

votes

**1**answer

88 views

### Covering points with a shortest lattice spiral

Let $S$ be a finite set of lattice points in $\mathbb{Z}^2$.
My question is, roughly:
Q. How can a shortest lattice spiral that passes through
every point of $S$ be found?
A lattice spiral (my ...

**9**

votes

**1**answer

231 views

### A random variation on Polya's orchard problem

Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, ...

**19**

votes

**5**answers

1k views

### What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...

**3**

votes

**5**answers

248 views

### Approaching convex and discrete geometry from other disciplines

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...

**0**

votes

**0**answers

115 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 ...

**2**

votes

**2**answers

318 views

### Practical use of estimates for the Gauss Circle Problem

This question is related to this and this ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere of radius $r$ centered at the origin. It is well known that $N(r) ...

**7**

votes

**1**answer

93 views

### Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?

Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of ...

**1**

vote

**1**answer

146 views

### Is there a “Bipartite” Szemeredi-Trotter theorem?

One version of the Szemeredi-Trotter theorem states the following:
Given a set of $L$ lines in the plane, the number of points incident to at least $k$ lines is bounded above by a constant times $L/k ...

**2**

votes

**2**answers

111 views

### Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...

**2**

votes

**1**answer

67 views

### Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...

**5**

votes

**2**answers

233 views

### Put P inside Q! polygons/polyhedra

We have two Polygons/Polyhedra P and Q.
Does there exist a polynomial time algorithm to decide if we can put P (using translation and rotation) inside Q or not?
First think about the case of which P ...

**10**

votes

**2**answers

491 views

### Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...