**4**

votes

**1**answer

433 views

### Algorithms for covering a rectilinear polygon using the same multiple rectangles

Sorry for the crossing-posting: original post is here
All angles of the polygon (representing a room) are right. It may be convex or concave. Use rectangles of the same size (representing a sensor ...

**7**

votes

**1**answer

302 views

### When is a 0-1 matrix a one-intersection incidence matrix?

The following problem is what motivated my previous MO question.
It is easily seen that for any given 0-1 matrix $M$, one can always find
a set $\mathcal P$ of points, and a set $\mathcal C$ of ...

**5**

votes

**0**answers

497 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 ...

**7**

votes

**0**answers

351 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 ...

**2**

votes

**1**answer

143 views

### Helly's Theorem for Biconvex Sets

Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...

**4**

votes

**1**answer

224 views

### convex polyhedron in the unit cube

Let $P$ be a given finite set of points within the $n$-dimensional unit cube. A finite set $Q$ of points within the $n$-dimensional unit cube covers $P$ if $\operatorname{conv}(Q) \supseteq P$ where ...

**1**

vote

**1**answer

119 views

### Deducing Linear Inequalities

Let $X_1,X_2,\ldots,X_n $ be indeterminates. Denote by $S$ the set of all linear inequalities of the form
$X_{i_1}+X_{i_2}+\ldots+X_{i_k} \geq k,$
with $k \in \{ 1,2,\ldots,n \}$ and $1 \leq i_1< ...

**4**

votes

**1**answer

205 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 ...

**10**

votes

**1**answer

314 views

### Needle probing for a convex body

Suppose there is an unknown closed convex body $K$ of
volume vol$(K) = V$ inside the
unit cube $[-\frac{1}{2}, \frac{1}{2}]^d$ in $\mathbb{R}^d$.
You are permitted to probe with a (one-dimensional)
...

**15**

votes

**2**answers

413 views

### Integer lattice points on a hypersphere

Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice ...

**9**

votes

**1**answer

390 views

### Question about tetrahedron decomposition

Are there tetrahedra which can be subdivided into three non-overlapping parts similar to the original? I believe this would require splitting one face into three parts. I know some types of tetrahedra ...

**2**

votes

**1**answer

153 views

### Is this cube packing possible?

I know how to pack $5$ unit squares in a square of side length $2+\frac{\sqrt{2}}{2}$. Is there an $\varepsilon>0$ such that there exists a packing of $9$ unit cubes in a cube of side length ...

**2**

votes

**1**answer

377 views

### Regularity of Delaunay triangulation of a hypercube

First using a three dimensional unit cube as an example for the term "regularity", we can have two possible triangulations:
(A)
(B)
We say the lower triangulation is more "regular" than upper ...

**3**

votes

**0**answers

91 views

### pavings and quadratic forms

Hi,
let $L$ be a lattice isomorphic to $\mathbb{Z}^r$ for some positive integer $r$ and $E=L\otimes \mathbb{R}$.
An integral paving in $E$ is a set $\Sigma$ of integral polytopes (the vertices are ...

**3**

votes

**1**answer

168 views

### Simplex with edges of length at least s having smallest circumradius

Is it true that of all $n$-simplices with edge lengths greater than or equal to some parameter $s$, the regular simplex with edge lengths $s$ has the smallest circumradius? It seems obvious, but I ...

**11**

votes

**1**answer

424 views

### Fano plane drawings: embedding PG(2,2) into the real plane

By a drawing of the Fano plane I mean a system of seven simple curves and
seven points in the real plane such that
every point lies on exactly three curves, and every curve contains
exactly three ...

**9**

votes

**1**answer

351 views

### Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...

**0**

votes

**1**answer

199 views

### Counting integer points in a Minkowski sum

We have known from Ehrhart theory that if $P$ is a $d$-dimensional polytope of $\mathbb R^n$ which has integer vertices then the number of integer points in $nP$ is a polynomial of degree $d$. We also ...

**2**

votes

**1**answer

132 views

### Realizability of extensions of a free oriented matroid by an independent set

Question:
I am searching for a non-realizable matroid with few dependencies relative to the number of points. Precisely, I would like to find a non-realizable (over $\mathbb{R}$) oriented matroid $M$ ...

**15**

votes

**1**answer

358 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...

**3**

votes

**1**answer

133 views

### The discrete theory of compressible fluids dynamics

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, ...

**12**

votes

**1**answer

447 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 ...

**5**

votes

**1**answer

514 views

### The Cayley Menger Theorem and integer matrices with row sum 2

I just filled a gap in my education by learning about the Cayley-Menger theorem, and the Cayley-Menger determinant:
If $P_0, \dots, P_n$ are $n+1$ point in $\mathbb{R}^n$, and $d_{i,j} = |P_i - P_j|$ ...

**2**

votes

**0**answers

85 views

### rigidity of isoradial graphs

Suppose given a $1$-separated net $\Gamma\subset\mathbb R^2$. Is it true or false that there exists $\delta>0$ and a $\delta$-isoradial graph containing $\Gamma$ as a subset of its vertices?
(I am ...

**30**

votes

**5**answers

859 views

### Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$,
except leave the origin $(0,0)$ unoccupied by a disk.
Q. Is it the case that every disk can be ...

**4**

votes

**3**answers

260 views

### Perimeter/Neighborhood of a graph on grid

Hello,
I have a $\sqrt{n}\times\sqrt{n}$ lattice graph $G=(V,E)$ i.e. vertices on said 2-dim integer lattice, and two vertices have an edge if their $L_1$ distance is one.
Now I want to claim ...

**2**

votes

**3**answers

494 views

### Strong notions of general position

Hi!
I am looking for notions of general position that are stronger than linear general position.
To illustrate, 3 points in linear general position don't lie on a line. I want a notion that would ...

**1**

vote

**0**answers

336 views

### 2d bin packing problem, with opportunity to optimize the size of the bin

I have been tasked with optimizing a manufacturing process. It is a non-trivial, NP hard problem. The problem is similar to the 2d bin packing problem, but we are trying to optimize the size of the ...

**2**

votes

**0**answers

94 views

### Symmetric dominance regions surrounding a Gaussian prime

Let $z=a + b i$ be a complex number which is a Gaussian prime,
on neither the $x$- nor the $y$-axis.
So $a^2+b^2$ is a prime.
Construct a region $D(z)$ surrounding $z$ which is the
largest ...

**2**

votes

**1**answer

197 views

### Maintaining boundary of unit circle arrangement

I have a process which in each step creates a new unit circle and I am interested in maintaining the boundary of the resulting arrangement in linear time.
Is there anything known about computing this ...

**4**

votes

**0**answers

129 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 ...

**3**

votes

**2**answers

345 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**

votes

**2**answers

259 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**

votes

**2**answers

787 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

**2**answers

182 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**

votes

**1**answer

167 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**

votes

**1**answer

176 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**

votes

**1**answer

212 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**

vote

**1**answer

404 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**

votes

**1**answer

255 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**

votes

**0**answers

457 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 ...

**9**

votes

**0**answers

130 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**

votes

**3**answers

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

**1**answer

377 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**

vote

**0**answers

128 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

**2**answers

225 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 ...

**27**

votes

**3**answers

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 $K$ ...

**26**

votes

**2**answers

897 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**

votes

**2**answers

563 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**

votes

**1**answer

748 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 ...