## Questions about Discs

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 1Consider a ball . Two players play the following game. The players alternate taking turns. On a player’s turn they must choose a single ball of fixed radius contained in 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 and 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 ? Does any one know what happens if you replace with a simplex?

Exercise 2Consider the closed ball of radius two with the Euclidean metric. Is there a constant such that any cover of by balls of radius one has a finite subcover of at most balls?

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

Exercise 3Let . Does there exist an such that 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 as 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 despite the fact that . However, the vertices of are a 2-net of size .

*Proof:* For certain 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 is going to be a circle of radius at most and hence area . 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 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:

and observe for small .

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