Can a finite collection of open discs cover a given set in two dimensions?

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  • Thread starter Ackbach
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    2016
In summary, a set in two dimensions is said to be "covered" by open discs when every point in the set is contained within at least one of the discs, leaving no gaps or uncovered areas. A finite collection of open discs cannot cover every set in two dimensions, as there are some complex shapes and sets that cannot be completely covered. The number of open discs needed to cover a set depends on its shape and complexity, but there is no specific formula for determining this number. Overlapping open discs can still cover a set in two dimensions, but their size and placement can greatly affect their efficiency in doing so. Smaller discs may require more to cover a set, while larger discs may have difficulty covering a more complex set. The placement
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Ackbach
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Here is this week's POTW:

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Let $\mathcal F$ be a finite collection of open discs in $\mathbb R^2$ whose union contains a set $E\subseteq \mathbb R^2$. Show that there is a pairwise disjoint subcollection $D_1,\ldots, D_n$ in $\mathcal F$ such that \[E\subseteq \cup_{j=1}^n 3D_j.\]
Here, if $D$ is the disc of radius $r$ and center $P$, then $3D$ is the disc of radius $3r$ and center $P$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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Re: Problem Of The Week # 241 - Nov 18, 2016

This was Problem A-5 in the 1998 William Lowell Putnam Mathematical Competition.

Honorable mention to Kiwi for a good attempt, but not quite correct. The solution, attributed to Kiran Kedlaya and his associates, follows:

Define the sequence $D_i$ by the following greedy algorithm: let $D_1$ be the disc of largest radius (breaking ties arbitrarily),
let $D_2$ be the disc of largest radius not meeting $D_1$, let $D_3$ be the disc of largest radius not meeting $D_1$ or $D_2$,
and so on, up to some final disc $D_n$. To see that $E \subseteq \cup_{j=1}^n 3D_j$, consider a point in $E$; if it lies in one of the $D_i$, we are done. Otherwise, it lies in a disc $D$ of radius $r$, which meets one of the $D_i$ having radius $s \geq r$ (this is the only reason a disc can be skipped in our algorithm). Thus the centers lie at a distance $t < s+r$, and so every point at distance less than $r$ from the center of $D$ lies at distance at most $r + t < 3s$ from the center of the corresponding $D_i$.
 

FAQ: Can a finite collection of open discs cover a given set in two dimensions?

What does it mean for a set to be "covered" by open discs in two dimensions?

When we say that a set in two dimensions is "covered" by open discs, it means that every point in the set is contained within at least one of the discs. In other words, the discs completely overlap with the set, leaving no gaps or uncovered areas.

Can a finite collection of open discs cover any set in two dimensions?

No, a finite collection of open discs cannot cover every set in two dimensions. There are some complex shapes and sets that cannot be completely covered by a finite number of discs, no matter how small or large they are.

How many open discs are needed to cover a given set in two dimensions?

The number of open discs needed to cover a set in two dimensions depends on the shape and complexity of the set. In general, the more complex the set is, the more open discs will be needed to cover it. However, there is no specific formula to determine this number as it can vary greatly.

Can a finite collection of open discs cover a set in two dimensions if the discs overlap?

Yes, a finite collection of open discs can still cover a set in two dimensions if the discs overlap. As long as every point in the set is contained within at least one of the discs, the set is considered to be covered. However, overlapping discs may not be the most efficient way to cover a set.

How does the size and placement of the open discs affect their ability to cover a set in two dimensions?

The size and placement of the open discs can greatly affect their ability to cover a set in two dimensions. Smaller discs may require more discs to cover a set, while larger discs may have difficulty covering a more complex set. The placement of the discs is also important, as overlapping or gaps between discs can impact their ability to cover the set efficiently.

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