**4**

votes

**2**answers

85 views

### Number of edges in linklessly embeddable graphs

Consider graphs over $n$ nodes.
What is the maximum number of edges of a linklessly embeddable graph?
A more general question is the following. Given $\mu$ what is the maximum number of edges of ...

**6**

votes

**1**answer

134 views

### Minimizing deep holes in sphere packings

What's the current state of knowledge regarding packings of spheres in $n$-space that minimize the supremum of the sizes of the holes? This notion of tightness is more rigid than asymptotic density. I ...

**5**

votes

**1**answer

111 views

### Ham sandwich theorem for discrete measures - reference request

A discrete version of the ham sandwich theorem states as follows (see for instance "Common Hyperplane Medians for Random Vectors" - Hill):
For every $\mu_1,...,\mu_n$ discrete (i.e., purely atomic) ...

**3**

votes

**1**answer

80 views

### Covering points with a shortest lattice spiral

Let $S$ be a finite set of lattice points in $\mathbb{Z}^2$.
My question is, roughly:
Q. How can a shortest lattice spiral that passes through
every point of $S$ be found?
A lattice spiral (my ...

**9**

votes

**1**answer

216 views

### A random variation on Polya's orchard problem

Polya's orchard problem is as follows:
"How thick must the
trunks of the trees in a regularly spaced circular orchard grow if they are
to block completely the view from the center?"
See, ...

**19**

votes

**5**answers

1k views

### What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...

**3**

votes

**5**answers

227 views

### Approaching convex and discrete geometry from other disciplines

I would like to learn some convex and discrete geometry (number 52 in MSC2010). I thought that it would be interesting to approach it from some other parts of mathematics - either by learning ...

**0**

votes

**0**answers

110 views

### “Open Points” in the 1983 proof of Szemerédi-Trotter theorem

I was reading through the 1983 paper "Extremal Problems in Discrete Geometry" and I was confused about the definition of "open point" appearing in this paper. By this point in the paper, the authors ...

**2**

votes

**2**answers

312 views

### Practical use of estimates for the Gauss Circle Problem

This question is related to this and this ones. The Gauss Circle problem asks for the number $N(r)$ of integer points within a sphere of radius $r$ centered at the origin. It is well known that $N(r) ...

**7**

votes

**1**answer

88 views

### Dropping altitudes to achieve nonobtuse planar triangulations: finite or infinite?

Given a planar triangulation of (say) a convex region,
imagine the following process to convert it to a triangulation with
no obtuse angles:
Pick an arbitrary obtuse angle at vertex $a$ of ...

**2**

votes

**1**answer

64 views

### Characterization of the medial axis of a surface

I would like to know if the following "characterization" of the medial axis of a surface is correct, and if so, how to prove it.
Let $S$ be a continuous, piecewise smooth, compact surface embedded in ...

**5**

votes

**1**answer

183 views

### Small remarkable matroids

Working on phased matroids (a generalization of oriented matroid to the complex case) I've found an interesting formula for computing the inner Tutte group (and, hence, all the Tutte groups) ...

**2**

votes

**0**answers

42 views

### Looking for N-dimensional spheres in the configuration space of the colorful Tverberg problem

Here we use standard notation for Tverberg's theorem: Dimension $d$, number of partition blocks $r$, and $N=(r-1)(d+1)$.
The configuration space of Tverberg's theorem is the simplicial complex ...

**2**

votes

**1**answer

122 views

### Regarding the set up of a geometry of numbers lemma

I have a question related to geometry of numbers, which although seems quite basic, I was rather confused by it so I decided to ask here. Let $\Lambda$ be a lattice in $\mathbb{R}^n$.
Let $R_1, ..., ...

**1**

vote

**0**answers

31 views

### Moment lemma for circular arc polygons

A "moment lemma" originally due to Fejes Tóth (I guess) states that if $P$ is any polygon (in $\mathbb{R}^2$) with $k$ sides, then for any $\mathbf{y}$, we have ...

**4**

votes

**1**answer

111 views

### The number of facets of a polyhedron under linear transformation

Consider a (not necessarily bounded) convex polyhedron $P\subset \mathbb{R}^n$ which has $k$ facets.
Let $L:\mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation.
Question1: Is there a fixed ...

**5**

votes

**1**answer

311 views

### Linear transformation of a polyhedron

Is there a simple proof that shows:
Linear transformation of a $\mathcal{H}$-polyhedron (i.e. the intersection of
finitely many closed half-spaces) is a $\mathcal{H}$-polyhedron.
Minkowski sum of ...

**8**

votes

**1**answer

401 views

### Polyhedron not circumscribed about a sphere

Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
My teacher ...

**11**

votes

**1**answer

313 views

### Longest of random worm-like paths in $\mathbb{Z}^2$

Imagine at each lattice point of $\mathbb{Z}^2$ within $[1,3n]^2$,
with coordinates
$\equiv 2 \mod 3$,
we place, with equal probability, one of these six patterns:
The result ...

**0**

votes

**0**answers

43 views

### Paths on Cartesian products of graphs satisfying linear constraints

Assume integers $d > r > 0$ and a connected graph $G$ with $d$ vertices. Every point on the $r$-fold Cartesian product of $G$ with itself, $G^{\square r}$, is equivalent to a dimension-$d$ ...

**5**

votes

**2**answers

99 views

### Discrete gradient on point clouds

I am interested to know some ways to approximate discrete gradient if you have a function on point clouds in 2D or 3D.
If you have a function defined on a grid, it well known that you can use a ...

**1**

vote

**1**answer

154 views

### Positroids and Totally Nonnegative Complex Grassmanian

Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case.
I found on arxiv the following interesting articles:
1)Alexander Postnikov: Total ...

**6**

votes

**1**answer

133 views

### Hiding $k$ disks inside a larger disk

Suppose one has $k$ unit-radius disks, and the goal is to hide them inside
a disk of radius $R \gg k$.
The detection probes are rays along a line.
(Think of the disks as tumor cells, and the rays as ...

**8**

votes

**1**answer

183 views

### Billiard dynamics with angle of reflection a fraction of angle of incidence

Suppose that a billiard ball bouncing in a unit square (or a lightray reflecting
in a mirrored square) has the property that the angle of reflection is a fraction
of the angle of incidence, rather ...

**9**

votes

**1**answer

266 views

### Are there irregular tilings by L-polyominoes?

I wonder if one can tile the plane with an order-$n$ L-polyomino
in a fundamentally irregular manner.
I seek help in defining what should constitute "irregular."
An L-polyomino of order $n \ge 2$ is ...

**0**

votes

**1**answer

55 views

### mean length of the non-crossing graphs on n points

My original question is rather vague so I'll start with a precise example and then indicate possible generalisations.
Given a n-tuple $x=(x_1,\dots,x_n)$ in, say, a square with side-length $1$ in the ...

**1**

vote

**0**answers

48 views

### Approximating Unit covering of d-dimensional points

Given a $d$-dimensional disk of radius $2$ in $\mathbb{R}^d$, how many disks of radius $1$ suffice to cover it. Of course, it's fine if the smaller disks overlap. What matters is to specify a finite ...

**1**

vote

**1**answer

72 views

### Unit covering of $d$-dimensional points

Given a set of points in $X$ axis, we want to cover them with minimum number of unit intervals.
For this problem we can assume that each interval in the optimal solution is starting or ending in one ...

**2**

votes

**1**answer

93 views

### Bound on maximum distance between points on a unit N-Sphere

I want to select M points on the N-sphere such that $min_{i\neq j,i,j\in \{1..M\}} ||x_i - x_j||$ is maximized.
Are there good upper bounds for this max-min distance?

**7**

votes

**2**answers

269 views

### Wait time to grid network disconnection with failing edges

Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node
connected to its four neighbors, with the top row connected to the bottom,
and the right column connected to the left.
Suppose ...

**2**

votes

**1**answer

110 views

### The Mahler conjecture and non-zonoidal 3-polytopes (4-polytopes)

I have been working on the Mahler conjecture for over a year now and have made some progress for certain classes of convex polytopes and I'm now attempting to write up my results specified to ...

**5**

votes

**1**answer

292 views

### A result from Peter McMullen's thesis

The classical definition of regular polytopes is recursive. It says that a polytope is regular if its facets and vertex figures (both smaller-dimensional polytopes) are regular.
The modern definition ...

**14**

votes

**0**answers

271 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 ...

**1**

vote

**1**answer

34 views

### Maximum crossings of curvature-constrained curve

Let $C$ be a curve in the plane whose curvature is everywhere $\le 1$.
If $C$ has length $L$, what is the largest number of proper self-crossings
of $C$ as a function of $L$?
For example, the curve ...

**1**

vote

**1**answer

129 views

### Calculate the discrete set of points B which are in the convex hull of the set of points A

This problem is likely best described with the following picture:
Given the discrete set of points $A$ (shown in blue), I wish to calculate the discrete set of points that are contained within the ...

**3**

votes

**3**answers

91 views

### Minimal area of non-planar lattice curves

Consider a $\mathbb{Z}^d$ lattice whose edges connect nearest-neighbor sites only, i.e. a $d$-dimensional hypercubic grid. Let $C$ be a closed curve along such edges. In general, for $d>2$ such ...

**27**

votes

**1**answer

895 views

### Can we find lattice polyhedra with faces of area 1,2,3,…?

I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...

**6**

votes

**1**answer

294 views

### Triangle (constrained number, rather than shape) packing?

Are there any interesting results on optimal packings in the plane using a fixed number of triangles (without a fixed size or shape constraint)?
For instance, what's the maximum area packing of the ...

**6**

votes

**1**answer

154 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

**1**

vote

**0**answers

43 views

### Euclidean embedding of a graph based on 1-ring neighborhood distances only

Consider a graph $(V,E)$, $\vert V \vert = n$ and weights $\{l_{ij}\}$, where $l_{ij}>0$ iff there is an edge connecting vertices $v_i$ and $v_j$. Distances beyond the 1-ring neighborhood are not ...

**4**

votes

**2**answers

123 views

### Computational complexity of deciding isomorphism of rational polyhedral cones

Let $C,C'$ be rational polyhedral cones in $\mathbb R^n$ both with non-empty interior. Rational means they are generated by vectors with rational entries. One says that $C,C'$ are isomorphic if there ...

**2**

votes

**1**answer

111 views

### Omit each vertex in turn of convex polygon: Iterative limit?

Let $P=P_0$ be a convex polygon of $n$ vertices $v_k$.
Let $P_{i+1}$ be the convex polygon obtained by intersecting the halfplanes
determined by the lines through every other vertex.
Below, $P_0$ is ...

**7**

votes

**2**answers

307 views

### get a point in polygon (maximize the distance from borders)

I have several 2D polygons represented by lists of xy-coordinates of their vertices.
It is needed to get several points inside the polygon so that they lie possibly far from the polygon's borders ...

**6**

votes

**0**answers

164 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 ...

**3**

votes

**1**answer

542 views

### The Stock Market Polytope: Explanation?

Ovidiu Racorean.
"Crossing Stocks and the Positive Grassmannian I: The Geometry behind Stock
Market."
(arXiv Abstract link)
Anyone care to offer a summary of what's going on here?
(The ...

**14**

votes

**2**answers

470 views

### Spearing rolling hula hoops

Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, ...

**1**

vote

**1**answer

352 views

### Study of convex polytopes via commutative algebra

Let $P \subset \mathbb{R}^d$ be any convex polytope with integral vertices, and let $M$ be the additive submonoid of $\mathbb{R}^{d+1}$ which is generated by $\{ (v,1) : v \in P \cap \mathbb{Z}^d \}$. ...

**18**

votes

**2**answers

364 views

### Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...

**5**

votes

**2**answers

199 views

### Genus of Tutte-Coxeter Graph

What is the genus of the Tutte-Coxeter graph -- the incidence graph of the
GQ of order 2? Seems like it should be well known, since nearly every other
parameter for that graph is known, but I can ...

**2**

votes

**1**answer

53 views

### Integer point in a non-empty polytope

I have a high-dimensional, non-empty polytope $Ax\geq b$ sitting inside the cube ($0\leq x_i \leq 1$). Is there any general theory or technique to show that this polytope contains an integer point, ...