# Tagged Questions

**3**

votes

**0**answers

141 views

### Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk
and a point directly above the disk center at height $h$.
Is the ...

**9**

votes

**0**answers

144 views

### Characterizing the norms on $\mathbb{R}^3$ coming from Platonic solids

Recall that any sufficiently nice compact centrally symmetric convex body in $B \subset \mathbb{R}^3$ gives rise to a Banach norm on $\mathbb{R}^3$ which has $B$ as its unit ball.
Is there a nice ...

**4**

votes

**1**answer

160 views

### Cover of a n-simplex with balls

Consider a n-simplex. For each edge (i,j), consider a n-ball, such that vertices i and j are antipodal on this ball. Is the simplex covered by the union of these balls? Thank you.

**7**

votes

**0**answers

329 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 ...

**11**

votes

**2**answers

468 views

### What is the largest possible thirteenth kissing sphere?

It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting. This was apparently the subject of disagreement ...

**13**

votes

**0**answers

467 views

### $\epsilon$-nets with respect to the cut norm

The cut norm $||A||\_C$ of a real matrix $A = (a_{i,j}) \in \mathcal{R}^{n\times n}$ is the maximum over all $I \subseteq [n], J \subseteq [n]$ of the quantity $\left|\sum_{i \in I, j \in ...

**4**

votes

**1**answer

356 views

### Are there non-tiling polyhedra that pack arbitrarily well?

The fact that an upper bound on the packing density $< 1$ has only recently been exhibited for regular tetrahedra in $\mathbb{R}^3$ (see this question) suggests that proving concrete bounds of ...

**11**

votes

**1**answer

778 views

### Cobounded ⇒ cocompact?

Assume $\Gamma$ acts by isometries on a separable Hilbert space $H$, and
$$\operatorname{diam} H/\Gamma\le 1.$$
Is it true that $H/\Gamma$ is compact?
Stupid example. Assume the action of ...

**9**

votes

**1**answer

393 views

### Are packing-homogeneous spaces homogeneous?

Given a metric space (M,d) define the packing function P(x,R,r) to be the maximum number of non-intersecting balls of radius r with centers in the ball B(x,R). Let’s call M packing-homogeneous if the ...

**13**

votes

**1**answer

2k views

### A circle packing conjecture

Consider $n$ circles with variable radii $r_1,\ldots, r_n$ that pack inside a fixed circle of unit radius. In other words, all $n$ variable-radius circles are contained in the unit radius circle and ...

**9**

votes

**1**answer

544 views

### Upper bound for tetrahedron packing?

There have been several recent advances on packing regular tetrahedra in $\mathbb{R}^3$. All the results I've seen have been lower bounds -- first John Conway and Sal Torquato showed that there ...

**7**

votes

**2**answers

478 views

### Coiling Rope in a Box: Decidable?

Is the problem Coiling Rope in a Box decidable? To be specific, is this decidable?
Given $L > 0$ and $r \in (0,\frac{1}{2})$,
both rational,
can a rope of length $L$ and radius $r$
fit ...

**17**

votes

**1**answer

1k views

### Coiling Rope in a Box

What is the longest rope length L of radius r that can fit into a box?
The rope is a smooth curve with a tubular
neighborhood of radius r, such that the rope does not
self-penetrate. For an open ...

**9**

votes

**1**answer

617 views

### Packing twelve spherical caps to maximize tangencies

Suppose that $v_i$, for $i \in \{1, 2, \ldots 11, 12\}$, are twelve unit length vectors
based at the origin in $R^3$. Suppose that $|v_i - v_j| \geq 1$ for all $i
\neq j$. What arrangement of ...

**16**

votes

**3**answers

2k views

### How many unit squares can you pack into a rectangle with nearly integer side lengths?

Earlier today, somebody asked what looks like a homework problem, but admits the following reading which I think is interesting:
Suppose $a_1,\dots, a_n$ are positive integers, and $\varepsilon$ ...