**35**

votes

**0**answers

557 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 {$L_1,\dots,L_m$} ...

**28**

votes

**0**answers

685 views

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**20**

votes

**0**answers

312 views

### Minimal number of intersections in a convex $n$-gon?

For a convex polygon $P$, draw all the diagonals of $P$ and consider the intersection points made by those diagonals. Let $f(n)$ be the minimal number of such intersections where $P$ ranges over all ...

**17**

votes

**0**answers

328 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 ...

**17**

votes

**0**answers

826 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial ...

**15**

votes

**0**answers

477 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 ...

**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

338 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 ...

**13**

votes

**0**answers

303 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 ...

**13**

votes

**0**answers

2k views

### Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes).
Is there an efficient way to calculate this?

**12**

votes

**0**answers

226 views

### Tiling a square with rectangles

Is it possible to completely tile a square with different rectangles of integer sides but all with the same area?
The original problem, not requiring integer sides for rectangles, was proposed by Joe ...

**12**

votes

**0**answers

402 views

### Reciprocity (Ehrhart-style) for real polytopes?

Is there some sense in which the well-known Ehrhart reciprocity law for rational, convex, polytopes can be extended to any convex polytope with arbitrary real vertices?
In other words, given any ...

**10**

votes

**0**answers

186 views

### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...

**10**

votes

**0**answers

319 views

### A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...

**9**

votes

**0**answers

253 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 ...

**9**

votes

**0**answers

134 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 ...

**9**

votes

**0**answers

325 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

175 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

604 views

### An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point:
Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with
the contravariant model structure (cofibrations are monomorphisms ...

**7**

votes

**0**answers

147 views

### Which -icial sets produce the “standard” representations of symmetric groups?

Suppose you have a system of cell complexes (say, even convex polyhedra) $(P_n)_{n\geqslant0}$ which occur as faces of each other and are used to define the corresponding notion of "$P_*$-set". So ...

**7**

votes

**0**answers

93 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 $k$ ...

**7**

votes

**0**answers

150 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 ...

**7**

votes

**0**answers

91 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

355 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

283 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 ...

**7**

votes

**0**answers

269 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

280 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

307 views

### Smoothing a piecewise smooth manifold

Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can ...

**6**

votes

**0**answers

135 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...

**6**

votes

**0**answers

86 views

### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...

**6**

votes

**0**answers

965 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

229 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 ...

**6**

votes

**0**answers

468 views

### Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...

**5**

votes

**0**answers

52 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 ...

**5**

votes

**0**answers

68 views

### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...

**5**

votes

**0**answers

128 views

### Is there a decomposition strengthening of the Sauer-Shelah Lemma?

Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates ...

**5**

votes

**0**answers

69 views

### Algorithm to express a point from a H-polyhedron as convex combination of extreme points

Let $P\subset\mathbb{R}^n$ be a convex polyhedron described as an intersection of hyperspaces, that is,
$$P:=\{\boldsymbol{x}: A\boldsymbol{x} \leq \boldsymbol{b}\}$$
Let $\boldsymbol{x} \in P$. We ...

**5**

votes

**0**answers

117 views

### Visibility in a prime orchard

This suggests a variant on Polya's orchard problem.
That problem asks1
for which radius $\epsilon$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the ...

**5**

votes

**0**answers

533 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

522 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 ...

**5**

votes

**0**answers

332 views

### $n$ lines in a general position and the number of empty triangles

Question. Consider $n \geq 5$ lines in a general position (i.e. no two lines are parallel and no triple intersections are allowed) in $\mathbb{R}^2$. Let $T(n)$ denote the maximal number of empty ...

**4**

votes

**0**answers

107 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 $P$ ...

**4**

votes

**0**answers

109 views

### Exotic line arrangements

I would like to discuss about the following problem. Hopefully, you will suggest me some ideas and bibliography.
At first I provide some basic definitions to set up the notation.
Let us consider a ...

**4**

votes

**0**answers

78 views

### Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...

**4**

votes

**0**answers

41 views

### How does one go from convexity to submodularity?

If I have a function which is convex in the hypercube, $[-1,1]^n$ then when would it imply that its restriction to $\{-1,1\}^n$ is submodular?
It would be helpful is someone can share some specific ...

**4**

votes

**0**answers

84 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 ...

**4**

votes

**0**answers

287 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

134 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 ...

**4**

votes

**0**answers

187 views

### Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following.
Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...

**4**

votes

**0**answers

144 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 ...