Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

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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 ...
7
votes
1answer
241 views

Convex Polyhedra Scissors Congruence Problem

I am currently writing a geometry paper "Rectifications of Convex Polyhedra" and I am confused to have discovered what appears to be a remarkable discrete geometric fact: Conjecture: Let $P$ be a ...
5
votes
3answers
127 views

Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...
7
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2answers
80 views

Generate polyhedra by collapsing vertices of a polyhedron

I am looking for basic information about the following idea: (I) Consider a square. By collapsing two adjacent vertices, we obtain a triangle. (II) Consider a three-dimensional cube. By collapsing a ...
2
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2answers
68 views

Basic question about discrete minimal surfaces

Let $P$ be a convex polygon with $n > 3$ vertices $v_1, \ldots, v_n \in \mathbb{R}^2$, let $x$ be a point in the interior of $P$, and let $u$ be a function with prescribed values at the vertices of ...
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0answers
49 views

Choosing the weights of a Voronoi diagram — is this function always the gradient of another function?

This question is related to the earlier question Weighted area of a Voronoi cell . As in that question, let $X = \{ x_1,\dots,x_n\} $ denote a set of $n$ points in the unit square $S = ...
6
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2answers
224 views

Existence of finite set of points in the revolving circles

Let $k$ and $n$ be two fixed integers. Let $C$ denotes the circle with radius $4n$ (in the plane $\mathbb{R}^2$). Suppose $\{C_1,C_2\}$ shows the set of two arbitrary tangent circles with radius $2n$ ...
14
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2answers
413 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 ...
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2answers
224 views

Cardinality of non-integer points in the translation of the Minkowski sum of convex hull.

Let $\operatorname{conv}(a_1,\ldots,a_m)$ denote the convex hull of $\{a_1,\ldots,a_m\}$. Let $\mathbb{Z}_+=\mathbb{N}\cup\{0\}$ and $\mathbb{Q}_+$ denotes the positive (inluding 0) rational numbers. ...
4
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0answers
44 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 ...
8
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1answer
208 views

What can we learn from the newly discovered monohedral convex pentagonal tiling?

Wikipedia: https://en.wikipedia.org/wiki/Pentagonal_tiling#Stein_.281985.29_and_Mann.2FMcLoud.2FVon_Derau_.282015.29 Media coverage: ...
4
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1answer
185 views

Smoothening a measure, II

There is an almost invisible, but significant difference between the question below and that recently answered by Boris Bukh. Given a probability measure $\mu$ supported on a finite set ...
2
votes
1answer
120 views

Smoothening a probability measure

Given a probability measure $\mu$ supported on a finite set $S\subset{\mathbb R}^2$, define $$ f(z):=\max\left\{\frac{\mu(x)+\mu(y)}2\colon \frac{x+y}2=z,\ x,y\in S \right\}, \ z\in{\mathbb ...
9
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1answer
618 views

infinite configuration of lines

I was looking at some random problems and questions I liked when I was in high school and I found this one which I still cannot prove. Does there exist a configuration of a countable number of ...
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2answers
1k views

What is the oriented Fano plane?

One way to remember the multiplication table of the octonions is to use the following diagram (which I got from John Baez's online paper): if $(e_i,e_j,e_k)$ is one of the lines listed according to ...
4
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0answers
92 views

Set of balls which the number of the ball intersects lines on the plane is bounded

Does there exist the set of balls(may be not disjoint) $X=\{B_i\subset\mathbb{R^2};i\in I\}$, satisfing following properties?(Note that the ball has a positive real radius) Let the set of all lines ...
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0answers
240 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 ...
3
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1answer
184 views

Blowing up spheres in a face centered cubic (fcc) packing geometry just enough to cover the volume of the lattice

Imagine I have an infinite lattice of spheres packed in a face centered cubic (fcc) lattice geometry which has the basis: $((-1, -1, 0), (1, -1, 0), (0, 1, -1))$. Here, provided that sphere-sphere ...
5
votes
1answer
110 views

How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...
4
votes
2answers
86 views

Inscribed parallelotope in a $d$-simplex

The problem setup is simple: Given a $d$-simplex $\Delta_d:=\{(x_1,\cdots,x_d):x_i\geq 0,\sum_i x_i\leq 1\}$, can we construct a finite sequence of parallelotope $A_i$ so that $\Delta_d=\cup_{i=1}^N ...
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0answers
503 views

Maximal disjoint hyperplanes

Assume a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ is given occupying a codimension $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these ...
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0answers
98 views

From Planar Graphs To Tangent Circles

I have a conjecture: "For each planar graph with vertices $V_1, V_2,\ldots, V_n$ there exist disjoint circles $w_1,w_2,\ldots,w_n$ in the plane, such that for every $i,j$, $w_i$ is tangent to $w_j$ ...
1
vote
1answer
121 views

Maximal $\pi/2$-separated subset of the sphere

A subset $A$ of a metric space is called $\varepsilon$-separated if $$dist(x,y)> \varepsilon \mbox{ for all } x\ne y\in A.$$ (Notice that the inequality in my definition is strict.) What is the ...
12
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1answer
442 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 ...
6
votes
1answer
403 views

Approximating a real by a ratio of primes

Let $x$ and $y$ be positive reals in $(0,1)$ with $x < y$ and $y-x =\epsilon$. I seek smallest primes $p$ and $q$ such that $$x \le \frac{p}{q} \le (x+\epsilon) = y \;.$$ Q. What upper bound ...
2
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1answer
79 views

The circle with minimal radius covering known finite set of points on a plane

Given some points on a plane, how to determine the circle with minimal radius covering all these points?
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0answers
110 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 ...
4
votes
1answer
92 views

How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?

This is the isoperimetric type question. We know that in $\mathbb{R}^d$, balls are the sets that minimize the isoperimetric ratio $\frac{S^{d}}{V^{d-1}}$, where $S$ is the surface area and $V$ is the ...
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1answer
198 views

Convex polyhedron and its Gauß-curvature [closed]

I have asked this question on MathSE and no one could give me an answer. So I'll post my question here. What I am trying to prove: A convex polyhedron has positive Gauß-Curvature at every vertex. ...
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0answers
44 views

Algebraic independence in normed spaces

A set of $n$ points in $\mathbb{R}^2$ is algebraically independent over $\mathbb{Q}$ if there is no polynomial dependency among the $2n$ coordinates. A result (Lemma 3.3) from "Globally linked pairs ...
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0answers
92 views

need clarification of a paper by Fejes Toth

I would like some clarification about the following from Fejes Toth's paper "A stability criterion to the moment theorem" The setup is: For each positive integer $n$, let $r(H_n)$ and $R(H_n)$ ...
6
votes
2answers
396 views

Embedding of planar graphs

I've recently come across the following lemma. Lemma (Valiant): A planar graph $G$ with maximum degree $4$ can be embedded in the plane using $O(|V|)$ area in such a way that its vertices are at ...
4
votes
1answer
127 views

Motzkin polynomials and enumeration of chord diagrams

On page 12 of the paper Enumeration of chord diagrams on many intervals and their non-orientable analogs" by Alexeev, Andersen, Penner, and Zograf is a list of polynomials which are a refinement of ...
20
votes
2answers
1k views

Erdős-Szekeres for first differences

The classical Erdős-Szekeres theorem says that any sequence of $n^2+1$ real numbers contains a monotonic $(n+1)$-term subsequence. Suppose, however, that we want to find a subsequence which is not ...
4
votes
1answer
109 views

Special case of Erdos Distance Problem in a plane

Erdos in his Distinct distance Problem in a plane conjectured that the minimal number of distinct distance determined by $n$ points in a plane be $g(n)$, $$g(n) \sim \frac{cn}{\sqrt{\log n}}$$ But ...
2
votes
1answer
138 views

Number of lines of symmetry of a set of lattice points

Given some finite $S\subseteq\mathbb R^2$, it is clearly possible for $S$ to have arbitrarily many lines of symmetry. However, it is not very clear if the same is necessarily true for subsets of ...
3
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2answers
313 views

Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?

Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...
3
votes
1answer
119 views

Do random triangulation edge-flips maintain randomness?

Let $S$ be a fixed set of $n$ points in the plane in general position. Let $T$ be a triangulation of $S$, (somehow) selected uniformly at random from all triangulations of $S$. (There are an ...
2
votes
1answer
100 views

Maximal neighbour-full partition of $\{0,1\}^n$

What is the largest complete minor of the $n$-dimensional hypercube? (which we call $k(n)$) Alternatively, what is the partition of $\{0,1\}^n$ with each set connected and neighboring each other that ...
11
votes
2answers
296 views

Double kissing problem

Consider two touching unit balls which will be called central balls. What is the maximum number $k$ of non-overlapping unit balls so that each ball touches as least one of two central balls? An easy ...
8
votes
4answers
267 views

Diameter of random segment intersection graph?

I have an even number of points $n$ randomly distributed (uniformly) in a disk. Then the points are randomly connected to form $n/2$ segments, a perfect matching. Finally, I form the intersection ...
12
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3answers
261 views

Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$. What is known about $V_n$? Is there a ...
4
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1answer
114 views

Best polygonal approximation to a polynomial $\pm$ c

Let a planar region $R$ be defined by the vertical range bounded by a polynomial $f(x) \pm c$ with $c>0$ a constant, and with $x$ varying between the smallest and largest roots of $f(x)$. For ...
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5answers
307 views

Packing obtuse vectors in $\mathbb{R}^d$

I came across this attractive theorem: Theorem. In $\mathbb{R}^d$, there can be at most $d+1$ vectors that form an obtuse angle with one another. This was proved1 as a corollary of a lemma about ...
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7answers
1k views

What's that shape? Inferring a 3D shape from random shadows

Let $P$ be a bounded, simply connected region of $\mathbb{R}^3$. $P$ could be a polyhedron, or a smooth shape, or an arbitrary shape; I'll assume below that $P$ is a (non-degenerate, perhaps ...
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3answers
5k views

Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \leq k } ...
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1answer
194 views

Ask the name of a combinatorial theorem

It is a classical theorem. For given integer $n \ge 1$, among ${n\choose{n/2}} = 2^{(1-o(1)n)}$ strings in the cube $\{0, 1\}^n$ with weights $n/2$, i.e., $n/2$ indices are 1, there are at least ...
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2answers
315 views

The most number of points that realize only $k$ distinct distances

For $k \ge 1$, let $f_d(k)$ be the largest possible number of points $p_i$ in $\mathbb{R}^d$ that determine at most $k$ distinct (Euclidean) distances, $\|p_i-p_j\|$. Example. For points in the plane ...
3
votes
1answer
239 views

On Dehn's infinitesimal rigidity theorem

Dehn's theorem states that any simplicial strictly convex polyedron P in Euclidean 3-space is infinitesimally rigid (that is, any non-trivial first order deformation of P induces a variation of its ...
4
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3answers
541 views

Number of Hyper-cube cuts

In how many ways a single hyperplane can cut a hypercube? Two "ways" are considered different, if the sets into which they divide vertices of the hypercube are different. So e.g. a line can cut ...