3
votes
0answers
134 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
0answers
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
1answer
157 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
0answers
327 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
2answers
467 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
0answers
463 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
1answer
354 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
1answer
776 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
1answer
388 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
1answer
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
1answer
543 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
2answers
477 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
1answer
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
1answer
616 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
3answers
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 $\varepsilon$ ...