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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

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  • $\begingroup$ 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? $\endgroup$ – Yixin Cao Apr 15 '12 at 19:54
  • $\begingroup$ 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. $\endgroup$ – Jeffε Apr 15 '12 at 19:56
  • $\begingroup$ @JɛffE yes, that is exactly what I am trying to say. I will edit the question to clarify. $\endgroup$ – Joe Apr 15 '12 at 22:48
  • $\begingroup$ @YixinCao I provided a figure which should hopefully clarify things. $\endgroup$ – Joe Apr 15 '12 at 23:01
  • $\begingroup$ @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. $\endgroup$ – Yixin Cao Apr 15 '12 at 23:25
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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.

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

ball-chain-packing

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

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