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

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1answer
189 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. ...
14
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0answers
119 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 ...
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0answers
34 views

Is there a non-polyhedral metric on a polyhedral surface? [closed]

I know of Alexandrov's Uniqueness Theorem and I know that the induced metric of any polyhedral surface is polyhedral but I was wondering if it was possible to define a metric on a polyhedral surface ...
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0answers
94 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 $w_j$ ...
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1answer
115 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
433 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
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1answer
391 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 ...
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1answer
170 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 ...
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1answer
72 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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108 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
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1answer
89 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
193 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. ...
0
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0answers
40 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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91 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
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2answers
388 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
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1answer
124 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 ...
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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 ...
3
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1answer
100 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
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1answer
130 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
306 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
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1answer
110 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
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1answer
93 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 ...
10
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2answers
286 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 ...
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4answers
261 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 ...
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3answers
259 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 ...
6
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1answer
156 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 $2n$ ...
4
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1answer
110 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 ...
6
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5answers
260 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 ...
21
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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 ...
39
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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
193 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 ...
11
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2answers
305 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
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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 ...
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3answers
534 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 ...
4
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2answers
244 views

Breaking a rectangle into smaller rectangles with small diagonals

Say I am given a rectangle with dimensions $a \times b$ and an integer $n$. I'd like to break this rectangle into $n$ smaller rectangles $R_i$, and I'd like to make the maximum diagonal of any of ...
3
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2answers
162 views

Examples of toric threefolds

I am looking for examples of smooth projective toric threefolds $\mathbb P_\Delta$ such that the rational polytope $\Delta$ has only pentagonal faces and hexagonal faces. I quickly searched for ...
4
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2answers
64 views

Expressing a convex Polytope as a sublevel set of a function

Given an n-dimensional polytope $P$ in $\mathbb R^n$, Given as a convex hull of a finite set of points, $S$ I would like to construct an expliict formula for a function $f\colon \mathbb R^n \to ...
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0answers
129 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ be the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$. Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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6answers
3k views
8
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5answers
767 views

covering by spherical caps

Consider the unit sphere $\mathbb{S}^d.$ Pick now some $\alpha$ (I am thinking of $\alpha \ll 1,$ but I don't know how germane this is). The question is: how many spherical caps of angular radius ...
2
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1answer
88 views

Epsilon-approximations of set systems with finite VC-dimension

ECorollary 6.9 in A Guide to NIP theories by Pierre Simon proves the following Theorem. For every positive integer $k$ and every positive real $\varepsilon$ there is an integer $n=n(k,\epsilon)$ ...
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1answer
223 views

Looking for reference or proof to some facts stated on Anand Pillay's book

In my current work I am using facts 2.1.11 and 2.1.12 from Anand Pillay's book Geometric Stability Theory. The facts are stated as follows: Fact 2.1.11. Let $(S,\mbox{cl})$ be a locally ...
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1answer
150 views

Enumerating Lattice points

Let $A \in \mathbb{R}^{d\times d}$ be an invertible matrix. Consider the set $$P_d := A\mathbb{Z}^d = \{A x| x \in \mathbb{Z}^d \} \subset \mathbb{R}^d$$. and $$ Q_d := [-1,1]^d.$$ I am interest in ...
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5answers
2k views

Do unit quaternions at vertices of a regular 4-simplex, one being 1, generate a free group?

Choose unit quaternions $q_0, q_1, q_2, q_3, q_4$ that form the vertices of a regular 4-simplex in the quaternions. Assume $q_0 = 1$. Let the other four generate a group via quaternion ...
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0answers
57 views

Presentation of the Rybnikov matroid

In this well celebrated work Gregory Rybnikov exhibit an example of two arrangements with the same underlying matroid, but with fundamental groups which are not isomorphic. This is a key ...
5
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1answer
141 views

Integer decomposition of dilated integral polytopes

For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$. Question: Given $d > n + 2$ is it true that $$ ...
59
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6answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
8
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2answers
467 views

When does every point in a polytope lie along a chord between its edges?

Consider the 3-simplex, or tetrahedron, in 3-space. Regardless of the positions of the vertices, every point in the simplex lies on a chord between two non-adjacent edges of the simplex. Or, ...
6
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1answer
435 views

Using mirrors to make a non-convex polygon visible from a fixed interior point

Take a point $A$ inside a non-convex polygon $P$. Is it always possible to place a finite set of mirrors given by straight segments (not necessarily along the boundary of $P$, any position inside $P$ ...
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3answers
179 views

Isometric imbedding of finite metric space into standards spaces [duplicate]

Is it true that any metric space consisting of $n$ points can be isometrically imbedded into $n-1$ dimensional Euclidean space? Hyperbolic space? (For $n=3$ this is true.) If not, what are ...