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

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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 ...
3
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2answers
298 views

Questions on Discrete Exterior Calculus in numerial computing

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics: (Discrete Exterious Calculus is the newly developed ...
2
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2answers
209 views

Primitive orthogonal vectors/Unimodular matrices

Primitive sets of vectors are very important in the theory of point lattices, since they constitute the sets of vectors that are part of a basis for the lattice. A set of integer vectors ...
11
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2answers
774 views

Access to a preprint by D. N. Verma

Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...
6
votes
2answers
172 views

Untangling entwined rigid chains in 3-space

I am interested in exploring the degree of "tangledness" of two rigid chains in space. A polygonal chain is a simple (non-self-intersecting) path of segments in $\mathbb{R}^3$, viewed as a rigid body. ...
3
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1answer
147 views

Triangulation of the surface determined by sampling two of its cross-sections

I have a data set that essentially looks like the picture below, i.e., it's given by sets of points in $\mathbb{R}^3$ that sample the cross-sections of a certain surface that in principle I do not ...
7
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0answers
141 views

Fractional Helly for more than one piercing

Fractional Helly Theorem says the following: For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...
3
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1answer
203 views

What properties does generalized Delaunay triangulation have?

Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
1
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1answer
300 views

Finding a point farthest away from $k$ points in a polygon

There are $k$ points placed inside a polygon and I am interested in finding a point inside the polygon (not necessarily on its boundary) who's minimum distance to any of the $k$ points is maximized. ...
2
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1answer
243 views

Ways to look at a polyhedral graph

Motivation There are at least three interpretations of an abstract polyhedral (= planar 3-vertex-connected) graph: the 1-skeleton of a convex polyhedron (when embedded into $\mathbb{R}^3$) a ...
12
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0answers
422 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 ...
8
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119 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 ...
18
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3answers
1k views

Research trends in geometry of numbers?

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...
2
votes
1answer
329 views

Apollonian gasket and the degree of convergence

Let $r_1,r_2\dots$ be the radii of Apollonian gasket. I would like to know for which values $\alpha$ we have $$\sum_{n=1}^\infty r_n^\alpha<\infty.$$ I know that if three circles $A$, $B$ and ...
1
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0answers
122 views

Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices? In particular: Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
2
votes
2answers
215 views

Is there a combinatorial analogue of the Kazdan Warner theorem?

First let me state a result of Kazdan and Warner Let $M$ be a compact orientable two dimensional manifold. Let $f:M \rightarrow \mathbb{R}$ be a function that has the same sign as the Euler ...
25
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3answers
1k views

Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

First let me state two known theorems. Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then $$ \int \frac{K}{2 \pi} dA = \chi (M) $$ where $K$ ...
23
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2answers
779 views

The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square? By "nonoverlapping" I mean: not sharing an interior point. By "touch" I mean: sharing a boundary point. ...
8
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2answers
558 views

A Problem about partitioning $S^2$

Question: Can the 2-dimensional sphere $S^2$ be partitioned into four nonempty sets such that every circle in $S^2$ passes through just three of these four sets? Here, "just three" means "exactly ...
21
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1answer
682 views

Sane bound on number of moves for Maker-Breaker game on $\mathbb R^2$ for $\{0,1,2,3,4\}$

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 ...
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2answers
252 views

packing disks tightly in the plane

Given a discrete point set $S$ in ${\bf R}^2$ with a specified base-point $p_0 \in S$, label the remaining points as $p_1, p_2, \dots$ in order of increasing distance from $p_0$ (with ties resolved ...
2
votes
2answers
227 views

What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of subsets in a subdivision of the set of $30$ points into subsets such that all the points in each subset are on the boundary of the ...
4
votes
1answer
411 views

Combinatorial geodesics

[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.] I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics ...
8
votes
1answer
388 views

Polyhedra that combinatorially shadow a sequence

Let $P$ be a polyhedron in $\mathbb{R}^3$. Say that $P$ combinatorially shadows a sequence of natural numbers $S$ if there is a continuous rotation of $P$ such that its orthogonal-projection shadows ...
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1answer
215 views

Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect ...
9
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2answers
343 views

On well separated point sets in the plane

Let us say that a finite set $A$ in the plane is $1$-separated if: 1) it has an even number of points; 2) no open ball of diameter $1$ contains more than $|A|/2$ points. For a $1$-separated set $A$ ...
14
votes
1answer
427 views

The optimal constant in Vitali covering lemma

Let me restate Vitali covering lemma. Let $\{B_i\}_{i\in F}$ be a finite collection of balls in the $\mathbb{R}^n$. Then there is $S\subset F$ such that the balls $\{B_i\}_{i\in S}$ are disjoint and ...
14
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2answers
716 views

Particles chasing one another around a circle

Two particles start out at random positions on a unit-circumference circle. Each has a random speed (distance per unit time) moving counterclockwise uniformly distributed within $[0,1]$. How long ...
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0answers
165 views

Lattice-point enumeration question involving linear combinations of matrices

I would like to know some references to learn more about an answer to this question, if there are any references: Let $A_1, \dots , A_m$ and $B$ be $n\times n$ symmetric matrices. Let $$S = \{(x_1, ...
2
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0answers
98 views

Rational viewing points in a polygon

We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside ...
6
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1answer
196 views

Seeing the vertices of a polygon with rational angles

Given any convex polygon in the plane, is it always possible to find a point $p$ in its interior such that when we draw the line segments from $p$ to each of its vertices, the angles formed at $p$ are ...
13
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1answer
325 views

The sparsest planar net that captures every unit segment

Let $\cal C = \lbrace C_i \rbrace$ be a collection of rectifiable curves in the plane with the property that every unit-length segment meets at least one curve in at least one point. Call such a ...
11
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0answers
379 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 ...
12
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1answer
571 views

The geometry of crinkled aluminum foil

I wonder if the geometry of crinkled aluminum foil has been studied?            The above is a photo of foil I flattened to reuse. It might be ...
4
votes
1answer
246 views

Best upper bound on rate for q-ary codes

Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch which states that the rate $R(\delta)$ corresponding to ...
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2answers
135 views

Disks Packing Variant

Usually disk packing problems require that no two disks of the packing intersect. Does anybody know if the problem has been studied when disks may intersect but they are not allowed to contain the ...
37
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8answers
3k views

A sudden smiley? :-)

This is a vague question, and I will no doubt be (properly!) chastised for posing it. I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which has the ...
2
votes
2answers
189 views

Is there a simple test to determine whether a polytope is integral?

It is known that any rational convex polytope expressed as $\{ x\in\mathbb{R}^d : Ax \ge b \}$, where $A\in\mathbb{Z}^{k\times d}$ and $b\in\mathbb{Z}^k$, can be written as the convex hull of finitely ...
9
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2answers
545 views

Discrete Morse function from smooth one

Suppose I have a smooth manifold $M$ and a smooth Morse function $f$ on $M$. Is there a standard way to replace $M$ with a cell complex and $f$ with a discrete Morse function such the resulting ...
0
votes
3answers
277 views

The symmetry group of $\mathbb Z^d$

Let $d \ge 1$, and consider the integer lattice $\mathbb Z^d$. This is a homogeneous space, in the manner of the Erlangan Programm. I would like to write $\mathbb Z^d = G / H$, where $G$ is the ...
5
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0answers
456 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 ...
6
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3answers
343 views

Herringbone partitions of regions and surfaces

Let $R \subset \mathbb{R}^2$ be a region of the plane bounded by a Jordan curve. The boundary $\partial R$ could be a polygon, or a smooth curve—there are variations depending upon boundary ...
3
votes
1answer
223 views

Empty convex polytopes for random point sets

I know of the famous results on the Erdős-Szekeres empty convex polygon problem in the plane (the Happy-Ending Problem), and I know that there are higher-dimensional extensions. A great source ...
4
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2answers
301 views

Realization spaces for regular convex polytopes

Q1. Are there convex polytopes combinatorially equivalent to each of the regular polytopes that are realized with integer vertex coordinates? ...
11
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1answer
332 views

Detecting a hidden convex body with line probes

Imagine that, somewhere inside an origin-centered, unit-radius sphere $S$ in $\mathbb{R}^3$, sits a convex body $K$ of volume vol$(K)=\alpha (\frac{4}{3} \pi)$, with $\alpha < 1$ the fraction of ...
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1answer
2k views

Get Largest Inscribed Rectangle of a Concave Polygon

I'm looking for an algorithm to find a set of largest inscribed rectangles of a concave polygon where each rectangle must be collinear with one of the edges of the polygon. In other words, I want to ...
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2answers
2k views

Covering a Polygon with Rectangles

I am tyring to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle. I thought about ...
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2answers
197 views

Triangular grid with 4 edges per vertex

I am trying to create a triangular grid/mesh for a rectangular domain in $\mathbb{R}^2$ with the property that each vertex is shared by (at most) four edges. Is this possible to accomplish?
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0answers
312 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 ...
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2answers
355 views

What collections of convex sets result in non-trivial uses of Helly's theorem?

Consider Helly Theorem, taken from notes by Igor Pak: Let $X_1, \dots, X_n \in {\mathbb{R}}^2$ be convex regions in the plane such that any triple interesects $X_i \cap X_j \cap X_k \neq 0$. Then ...