# Counting the number of thick regions which overlap a square

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

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 .

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