<p>I think that the maximum number of pairwise disjoint fat regions which overlap the square should be strongly related to circle packing. </p>

<p>The worst-case shape for a region is something like a "ball &amp; chain". Below I have depicted such a region for $\beta=2$ with diameter 1</p>

<p><img src="https://i.sstatic.net/o8Gen.png" alt="ball-chain">.</p>

<p>and these can pack within distance 1 of the unit square obviously much more tightly than I've depicted them. </p>

<p><img src="https://i.sstatic.net/LMAaS.png" alt="ball-chain-packing"></p>

<p>Note that the actual ball &amp; chain region is defined by the green area, and the outer circle is just a guide to depict the fact that these regions have fatness 2. In fact, the chain part of the region, can "bend" to allow more regions to be packed.</p>

<p><img src="https://i.sstatic.net/2gYW4.png" alt="enter image description here"></p>

CC BY-SA 3.0
2012-04-16T20:37:33.373Z
2012-04-16T20:37:33.373Z
Joe
<p>Let $S$ be a unit square.
<strong>As a function of $\beta$, what is the maximum number of $\beta$-fat pairwise-disjoint regions <em>with diameter at least 1</em> which can intersect $S$?</strong> </p>

<p>Below, we give a figure showing that for $\beta=1$, the maximum number is 7. What about for $\beta = 2, 3, \ldots, n$? </p>

<p>Recall the definition of <em>fat</em> for regions in the plane. Given a region $R$, let circle $C_1$ of radius $r_1$ be the largest circle contained in $R$, and let circle $C_2$ of radius $r_2$ be the smallest circle that contains $R$. The <em>fatness</em> of $R$ is given by $\frac{r_2}{r_1}$, and we say that $R$ is $\beta$-fat, for $\beta = \frac{r_2}{r_1}$. </p>

<p>For example, if $r_2 = r_1=\frac{1}{2}$, then the regions are unit circles, and there are at 7 circles with diameter at least 1 which can overlap $S$ without overlapping each other.
In the figure below, we have depicted a unit square and 7 unit circles which overlap the square.</p>

<p><img src="https://i.sstatic.net/ggOn5.png" alt="overlapping circles"></p>

CC BY-SA 3.0
2012-04-15T19:54:35.277Z
The condition "circles at least as large as $S$" is confusing, and if you are talking about areas, a circle of radius $1$ is not as large as $S$.  Also, for the $r_2 = r_1 = 1$ case, you can put $7$ circles (one in the middle of $S$), am I stupidly wrong?
Yixin Cao
CC BY-SA 3.0
2012-04-15T19:56:25.603Z
Your definition of "thick" is one of the standard definitions of "fat".  I assume you mean "the maximum number of thick disjoint regions **with diameter at least 1** that can intersect S", since otherwise there is no upper bound.  Tiny circles have thickness 1.
Jeffε
CC BY-SA 3.0
2012-04-15T22:48:19.513Z
@JɛﬀE yes, that is exactly what I am trying to say. I will edit the question to clarify.
Joe
CC BY-SA 3.0
2012-04-15T23:01:19.967Z
@YixinCao I provided a figure which should hopefully clarify things.
Joe
CC BY-SA 3.0
2012-04-15T23:25:15.037Z
@Joe As my picture shows, seven circles are possible.  The point is: two circles (almost) tangent to two opposite points.  My drawing is always bad, but I hope the graph is helpful.
Yixin Cao
CC BY-SA 3.0
2012-04-16T07:00:03.227Z
@Joe: If you rotate the circles slightly around the center of the square, all seven circles intersect the square.
Jeffε
CC BY-SA 3.0
2012-04-16T19:10:55.213Z
@YixinCao I don't see your figure, but I'm drawing up another one that pictures what you and J$\epsilon$ffe pointed out
Joe
CC BY-SA 3.0
2012-04-16T19:28:19.137Z
@JɛﬀE I see what you and YixinCao were saying, and I corrected the question.
Joe
<cg.comp-geom>
Counting the number of thick regions which overlap a square
Joe
# Counting the number of thick regions which overlap a square

(Asked by: username_0 on 2012-04-15)

Let $S$ be a unit square.
**As a function of $\beta$, what is the maximum number of $\beta$-fat pairwise-disjoint regions *with diameter at least 1* which can intersect $S$?**

Below, we give a figure showing that for $\beta=1$, the maximum number is 7. What about for $\beta = 2, 3, \ldots, n$?

Recall the definition of *fat* for regions in the plane. Given a region $R$, let circle $C\_1$ of radius $r\_1$ be the largest circle contained in $R$, and let circle $C\_2$ of radius $r\_2$ be the smallest circle that contains $R$. The *fatness* of $R$ is given by $\frac{r\_2}{r\_1}$, and we say that $R$ is $\beta$-fat, for $\beta = \frac{r\_2}{r\_1}$.

For example, if $r\_2 = r\_1=\frac{1}{2}$, then the regions are unit circles, and there are at 7 circles with diameter at least 1 which can overlap $S$ without overlapping each other.
In the figure below, we have depicted a unit square and 7 unit circles which overlap the square.

![overlapping circles](https://i.sstatic.net/ggOn5.png)

**Tags:** cg.comp-geom
> **Comments:**
> **username_1**: The condition "circles at least as large as $S$" is confusing, and if you are talking about areas, a circle of radius $1$ is not as large as $S$.  Also, for the $r_2 = r_1 = 1$ case, you can put $7$ circles (one in the middle of $S$), am I stupidly wrong?  
> **username_2**: Your definition of "thick" is one of the standard definitions of "fat".  I assume you mean "the maximum number of thick disjoint regions **with diameter at least 1** that can intersect S", since otherwise there is no upper bound.  Tiny circles have thickness 1.  
> **username_0**: @JɛﬀE yes, that is exactly what I am trying to say. I will edit the question to clarify.  
> **username_0**: @YixinCao I provided a figure which should hopefully clarify things.  
> **username_1**: @Joe As my picture shows, seven circles are possible.  The point is: two circles (almost) tangent to two opposite points.  My drawing is always bad, but I hope the graph is helpful.  
> **username_2**: @Joe: If you rotate the circles slightly around the center of the square, all seven circles intersect the square.  
> **username_0**: @YixinCao I don't see your figure, but I'm drawing up another one that pictures what you and J$\epsilon$ffe pointed out  
> **username_0**: @JɛﬀE I see what you and YixinCao were saying, and I corrected the question.  

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## 1 answers

### Answer by username_0 on 2012-04-16. Score: 2 | Accepted answer

I think that the maximum number of pairwise disjoint fat regions which overlap the square should be strongly related to circle packing.

The worst-case shape for a region is something like a "ball & chain". Below I have depicted such a region for $\beta=2$ with diameter 1

![ball-chain](https://i.sstatic.net/o8Gen.png).

and these can pack within distance 1 of the unit square obviously much more tightly than I've depicted them.

![ball-chain-packing](https://i.sstatic.net/LMAaS.png)

Note that the actual ball & chain region is defined by the green area, and the outer circle is just a guide to depict the fact that these regions have fatness 2. In fact, the chain part of the region, can "bend" to allow more regions to be packed.

![enter image description here](https://i.sstatic.net/2gYW4.png)


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