**0**

votes

**0**answers

7 views

### What is the 4D axis of rotation for Necker cube inversion?

See the figure on top of page 47 of Rudy Rucker's book. ...

**2**

votes

**2**answers

142 views

### Number of ways of tiling a $2 \times n$ rectangle using rectangles with integer sides

How many ways are there of tiling a $2 \times n$ rectangles using rectangular tiles with positive integer side lengths?
I've done some work on this and have found a way of calculating this that's ...

**4**

votes

**0**answers

112 views

### Reference for the notion of polyhedra “degenerations”

Let $P$ be a convex polyhedron and let $P(t)$ be a continuous deformation thereof, such that:
a) $P(0)=P$;
b) for all $t\in[0;1)$ the polyhedron $P(t)$ is strongly combinatorially equivalent to $P$ ...

**0**

votes

**0**answers

21 views

### Non-adjacent Pair of Edges with Minimal Weight Sum

Given an weighted, undirected Graph $G(V,E)$ without loops or parallel edges,
what is the complexity of determining a pair of non-adjacent edges, whose sum of weights is w.l.o.g. minimal?
...

**2**

votes

**1**answer

79 views

### Orthogonal embeddings and edge lengths

I'm interested in orthogonal embeddings of graphs into the 2-dimensional, i.e where vertices are placed at integer co-ordinates and edges are routed along the grid lines and are not allowed to ...

**2**

votes

**0**answers

138 views

### Polynomials representing locally constant functions

Let $K$ be a finite field with $p$ elements.
(a) Let $f\in K\lbrack x\rbrack$ be such that (i) $\deg(f)<p$ and (ii) $f(2x) = f(x)$ for $\geq (1-\epsilon) p$ values of $x$ in $K$. What can we say ...

**2**

votes

**0**answers

48 views

### Sharpening the Loomis-Whitney inequality

The Loomis-Whitney inequality implies that if $A\subset\mathbb Z^n$ is a finite, non-empty set of size $K:=|A|$, then, denoting by $K_1,\dotsc,K_n$ the sizes of the projections of $A$ onto the ...

**0**

votes

**0**answers

30 views

### Question on solving an optimization problem using Variable splitting and ADMM

Tell me if I have found the right approach to the following optimization problem:
$$
min_{x} \frac{1}{2}\left \| Ax-b \right \|_2^2
\\
s.t. \ \ \Phi v=x \ , \ {x^T(1-x)}=0
$$
$A$ and $\Phi$ ...

**3**

votes

**0**answers

71 views

### Is there a Havel-Hakimi for geometric graphs?

Suppose that we are given $n$ points in the plane, with a degree prescribed for each, and the question is whether we can place a geometric graph on them. Is there an efficient algorithm for this?
...

**1**

vote

**0**answers

54 views

### Building an orthogonal embedding for a 4-planar graph

I'm interested in the following paper http://www.computer.org/csdl/trans/tc/1981/02/06312176.pdf
In particular i'm interested in the construction Valiant describes to prove that it is possible to ...

**5**

votes

**5**answers

211 views

### Locked convex polyhedra

Call a set of polyhedra free if it is possible to rigidly move the polyhedra, without any polyhedron intersecting any other, so that their pairwise distances are arbitrary large, and locked otherwise. ...

**5**

votes

**0**answers

52 views

### convex hull of all-ones principal submatrices

For a subset $S$ of $\{1,\ldots,n\}$,
let $\mathbf{1}_S\in\{0,1\}^n$ denote the indicator vector of $S$, with a $1$ on the $i$th coordinate iff $i\in S$. Let $\mathcal{X}$ denote the convex-hull of ...

**2**

votes

**0**answers

51 views

### Characterizing subgroups of R^n with dense factors

It is well known that (additive) subgroups of $\mathbb{R}^n$ are products of discrete subgroups (lattices) by dense subgroups in subspaces. My question is the following: given a generator set of $p$ ...

**3**

votes

**2**answers

105 views

### How many different integer polytopes does square lattice have?

Let $E_n = \{ (i,j) : 0 \leq i,j \leq n-1 \}$. We say that a polytope $P$ is an integer polytope in $E_n$ if all vertices of $P$ belongs to $E_n$.
My question is how many different integer polytopes ...

**12**

votes

**1**answer

335 views

### On convergence of convex bodies

Let $K\subset \mathbb{R}^n$ be a compact convex set of full dimension. Assume that $0\in \partial K$.
Question 1. Is it true that there exists $\varepsilon_0>0$ such that for any ...

**0**

votes

**0**answers

16 views

### min-max problems about sets of points

Many research problems in discrete geometry have the following form:
Given a set $S_n$ of $n$ points, let $Max(S_n)$ be the maximum
[something]. Define:
$$MinMax(n) = \inf_{S_n} Max(S_n)$$ ...

**3**

votes

**0**answers

72 views

### Algorithm for finding the fewest cartesian products to partition a set of points

The question Algorithm for finding the fewest rectangles to cover a set of rectangles was already answered here and a similar question was answered here. Those questions were about regular geometric ...

**2**

votes

**1**answer

89 views

### What is the maximal diameter of a cell in a particular partition of the simplex?

Consider a standard simplex with points $(p_1, \dots, p_n)$, $p_i \ge 0$, and $\sum_i p_i = 1$. Fix a set $\{q_k\}_{k=1}^K$ with $0\leq q_k \leq \infty$ and $i,j\in\{1, \dots, n\}$. Partition it via ...

**6**

votes

**1**answer

169 views

### Exotic line arrangements

I would like to discuss about the following problem. Hopefully, you will suggest me some ideas and bibliography.
At first I provide some basic definitions to set up the notation.
Let us consider a ...

**10**

votes

**1**answer

215 views

### Optimization of points on a plane

Suppose we have $n$ points on a plane. Let $D$ be the sum of the squares of all the pairwise distances between the points. Let $A$ be the area of the convex hull. What is the minimum possible value of ...

**2**

votes

**1**answer

106 views

### Is it possible to cover all pairs of points at distance at most 1 by constant number of partitions into sets of diameter at most 1?

Let $n$ be a natural number and let $S_n$ be a square $[0,n] \times [0,n]$ in the plane.
We say that a partition $\mathcal{Q} = R_1 \cup \cdots \cup R_t$ of $S_n$ is simple if each of the sets $R_1, ...

**1**

vote

**0**answers

42 views

### VC dimension of infinite cones [closed]

What is the VC dimension of infinite $d$-dimensional cones? ( single cones not double).
I would say $2d + 1$ or $O(d^2)$
Does anybody have any reference or ideas?

**0**

votes

**0**answers

19 views

### Is there a shelling of a (threshold)shifted complex, such that any partial shelling is still (threshold)shifted?

first the relevant definitions:
A complex $\Delta \subset 2^{[n]}$ is a family of subsets of $[n]$ that is closed downwards, i.e. if $A \subset B$ and $B \in \Delta$, then $A \in \Delta$.
A complex ...

**0**

votes

**0**answers

55 views

### What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?

I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...

**7**

votes

**1**answer

192 views

### Unusual isoperimetry and maximizing the measure of unions of translates of a set

Let me state a standard result first. Let a $A\subset \mathbb{R}^d$ be a set of fixed volume. Define $A_t$ to be the set of all points at distance at most $t$ from $A$. Then the volume of $A_t$ is ...

**5**

votes

**2**answers

265 views

### Conjugate transpose and discreteness, for Kleinian groups

Let $G=\langle g_1,\dots g_n\rangle<\mathrm{PSL}_2(\mathbb{C})$ be discrete,
i.e. a finitely generated Kleinian group.
Let $H=\langle g^{\dagger}g\mid g\in G\rangle$,
(the group generated by the ...

**4**

votes

**1**answer

268 views

### Generating function for number of different tessellation checkered rectangle

Let $R_n$ be checkered rectangle sized $n \times 4, n \ge 1$.
Let $a_n$ be number of different $R_n$ tiling with rectangles sized $1 \times 3$.
$\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ $ $\ \ \ ...

**5**

votes

**0**answers

68 views

### What is the maximal convex hull in $\mathbb R^3$ of a tree with fixed total length?

Denote by $\mathcal T_n$ the set of all trees on $n$ nodes. For a tree $T\in\mathcal T_n$, we assign to each edge a non-negative length such that the sum of all lengths is 1. Denote by $v(T)$ the ...

**0**

votes

**0**answers

107 views

### The smallest disk containing all cirular arcs

In a comment to my recent question about covering segments by a disk, Gerhard Paseman has suggested a generalisation: replacing the segments of the original $n$-gon by a simple closed (say, convex) ...

**6**

votes

**0**answers

307 views

### Smoothing a piecewise smooth manifold

Let $M \subset \mathbb{R}^d$ be a piecewise smooth $2$-manifold. Let $C$ be a polyhedral complex that covers $\mathbb{R}^d$ and contains faces of dimension $[0,d]$. Since $M$ is a $2$-manifold, we can ...

**0**

votes

**0**answers

23 views

### Efficient sampling from a polytope with large number of contraints [duplicate]

As far as I know, the most popular way to sample from a polytope (in H-representation)
\begin{equation}
\mathcal{P} := \{z \in \mathbb{R}^n | (Az)_j \le b_j\; \forall j=1,2,\ldots,m\}
\end{equation}
...

**11**

votes

**1**answer

171 views

### Is there a proof of the uniformization theorem using circle packing?

In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...

**5**

votes

**1**answer

131 views

### Are the primary parallelotopes classified? (equivalently, Voronoi cells of lattices)

A primary parallelohedron is a polyhedron that can fill space with infinite translated copies.
It is known (e.g., Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, pp. 29-30, 1973; or, ...

**1**

vote

**0**answers

42 views

### Number of simplices contained in a convex body

I am interested in the following question:
Given a convex body $K$ in $\mathbb{R}^d$ and an $\epsilon>0$ small enough, how many $d$-simplices $\{D_i\}_{i=1}^m\subset K$ do we need at least so that ...

**0**

votes

**1**answer

58 views

### A bound on the Haussdorff distance

Let $X, Y \subset \mathbb{Z}^2$ be two discrete and bounded sets. Let $f_X$ be the Euclidean signed distance function of $X$ (similarly for $Y$) and $d_H(X,Y)$ the Euclidean Haussdorff distance ...

**9**

votes

**3**answers

256 views

### Labeling edges of an icosahedron with sum constraints

The question is inspired by this previous MO question. There it was shown that it's possible to label the edges of a cube by the numbers $\{1,2,\ldots,6,8,9, \ldots, 13\}$ in such a way that:
Three ...

**3**

votes

**1**answer

182 views

### Status of Zeeman's collapsability Conjecture

Zeeman's conjecture in topological combinatorics states that if K is a contractible polyhedron of dimension 2, then K×I has a collapsible subdivision.
What is the status of this conjecture ...

**15**

votes

**5**answers

587 views

### The smallest disk containing all sides of an $n$-gon

Start with a regular $n$-gon of side 1 and consider its sides as open segments that can be moved around in the plane, allowing only translations. Two segments may not intersect.
What is the ...

**0**

votes

**0**answers

59 views

### Maximum value of linear function on the intersection of a parametrized family of balls

Let $C$ be a (nonempty) closed convex subset of $\mathbb{R}^n$ and $a, b \in \mathbb{R}^n$. Using the normal cone characterization of the euclidean projection operator $\mathrm{proj}_C$ (recall that ...

**3**

votes

**1**answer

100 views

### How to show it is contained in a convex hull?

There are $(d+1)f$ points(denote set of all points as $S$) in $\mathbb{R}^d$, that can be divide into $d+1$ disjoint sets $F_1,...,F_{d+1}$, each set of size $f$. If we have
$$
\mathcal{H}(F_i)\bigcap ...

**19**

votes

**1**answer

623 views

### Covering of a surface of a cube $n\times n \times n$ by pieces of paper $1\times 6$

When I was too young one of my problems was in the list of problems of All-Russian Olympiad. The problem is the following:
Problem. We have a surface of a cube $n\times n \times n$ such that each ...

**1**

vote

**0**answers

62 views

### Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb ...

**5**

votes

**1**answer

154 views

### Intersection of rotating regular polygons

This question has a recreational flavor, but may not be
entirely uninteresting.
Let $P_k$ be a unit-radius regular polygon of $k$ sides,
and $P_n$ a unit-radius regular polygon of $n \ge k$ sides.
...

**4**

votes

**0**answers

78 views

### Finding closest set of K disjoint hyperspheres to a point in $\mathbb{R}^n$ with uniform radius

I am interested in the following problem: in $\mathbb{R}^n$, we have $N$ overlapping hyperspheres all with the same radius. Given a point $p$ in $\mathbb{R}^n$, the objective is to find the $K$ non ...

**6**

votes

**1**answer

85 views

### A question about simple closed plane polygons

For any positive integer $n$ greater than $3$, let $P(1),P(2),...,P(n)$ be a set of $n$ pairwise distinct points in the Euclidean plane, no three of which are collinear. Let $H(P(1),P(2),...,P(n))$ be ...

**13**

votes

**0**answers

229 views

### Tiling a square with rectangles

Is it possible to completely tile a square with different rectangles of integer sides but all with the same area?
The original problem, not requiring integer sides for rectangles, was proposed by Joe ...

**5**

votes

**0**answers

129 views

### Is there a decomposition strengthening of the Sauer-Shelah Lemma?

Let $S \subset \{-1,1\}^n$. For a subset $A \subset [n]$ let $P_A$ denote the coordinate projection operator on S; in other words let $P_A(S)$ be the coordinate projection of $S$ onto the coordinates ...

**14**

votes

**1**answer

275 views

### separating points in $\mathbb{R}^d$ by minimal number of planes

Given $n$ points of general position in $\mathbb{R}^d$ (say, $n>d$ and no $d+1$ lie in a hyperplane.) We want to draw $k$ hyperplanes not passing through those points so that they all are in ...

**14**

votes

**1**answer

297 views

### Number of height-limited rational points on a circle

Consider origin-centered circles $C(r)$ of radius $r \le 1$.
I am seeking to learn how many rational points might lie on $C(r)$,
where each rational point coordinate has height $\le h$.
For example, ...

**7**

votes

**3**answers

225 views

### Embedding planar graphs into the grid

I've seen the following lemma in a paper. The result is by 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 ...