Parker Glynn-Adey

Questions about Discs

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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