Posted in Math by pgadey on 2013/07/20

Over the past couple weeks I’ve been asked a lot of questions about discs in Euclidean space. In this post we’ll be putting pennies on a table, refining covers of discs, and trying to cram lots of balls into high dimensional balls. Some open questions about putting pennies on tables occur below.

Exercise 1 Consider a ball ${B = B_R(0) \subset \mathbf{R}^N}$. Two players play the following game. The players alternate taking turns. On a player’s turn they must choose a single ball ${B_r(x_i) \subset B}$ of fixed radius ${r}$ contained in ${B}$ which is disjoint from all previously chosen balls. A player loses when they cannot pick such a ball. Does either player have a winning strategy?

If ${n=2}$ and ${r < R}$ then one can interpret this game as players putting pennies on a circular table top. The first person to be unable to place a penny on the table losses.

What happens if one varies ${B}$? Does any one know what happens if you replace ${B}$ with a simplex?

Exercise 2 Consider the closed ball of radius two ${B = \bar{B}_2(0) \subset \mathbf{R}^2}$ with the Euclidean metric. Is there a constant ${n}$ such that any cover of ${B}$ by balls of radius one has a finite subcover of at most ${n}$ balls?

Ben Green asked this question to his analysis students as a homework assignment with ${B}$ replaced by an arbitrary compact metric space. Mike Pawliuk dug it up while he was searching for point-set topology questions.

Exercise 3 Let ${\epsilon > 0}$. Does there exist an ${N \in \mathbb{N}}$ such that ${B_{1 + \epsilon}(0) \subset \mathbf{R}^N}$ contains a 2-net with more than a hundred points?

Mike Pawliuk also asked me this question in connection with his point set-topology course. The interesting part here is that ${\textrm{Vol}_N(B_{1+\epsilon}(0)) \rightarrow \infty}$ as ${N \rightarrow \infty}$ but this is not enough to gaurantee a packing. The usual example of this is the fact that a single unit ball fits into the cube ${C_N = [-1,1]^N}$ despite the fact that ${\textrm{Vol}_N C_N = 2^N}$. However, the vertices of ${C_N}$ are a 2-net of size ${2^N}$.

Proof: For certain ${\epsilon}$ there is no such net. We show that there is no net with more than two points. Take a plane which contains three points of the net. The intersection of this plane with ${B_{1+\epsilon}}$ is going to be a circle of radius at most ${1+\epsilon}$ and hence area ${\pi(1+\epsilon)^2}$. Now we compute the smallest area of a metric one ball inside this circle. The smallest such an area could be is if we are looking at two unit balls in the plane since increasing ${\epsilon}$ will increase the size of the smallest possible intersection with a unit ball. The smallest possible intersection will occur when the metric ball is centered on the boundary of the unit ball. In two dimensions we’ll get a vesica piscis. We then compute the area:

$\displaystyle A = \textrm{Vol}_2 \left(B_1\left(\left(0,\frac{ 1 }{ 2 }\right)\right) \cap B_1\left(\left(0,-\frac{ 1 }{ 2 }\right)\right)\right) = \frac{ 2 \pi }{ 3 } - \frac{ \sqrt{3} }{ 2 }$

and observe ${3A > \pi \cdot (1+\epsilon)^2}$ for small ${\epsilon}$. $\Box$

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