Return to Question

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 simplicity, let $r_1=1$, and let $2$ be the side length of a square $S$.

I want to know the maximum number of $\frac{r_2}{r_1}$-fat pairwise-disjoint regions with diameter at least 1 which can intersect $S$.

For example, if $r_2 = r_1=1$, then the regions are circles, and there are at most 6 circles with diameter at least 1 which can overlap $S$ without overlapping each other. In the figure below, we have depicted a square and 7 circles, but the circles above and below the square cannot overlap the square without also overlapping the circle in the center of the 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.

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 simplicity, let $r_1=1$, and let $2$ be the side length of a square $S$.

I want to know the maximum number of $\frac{r_2}{r_1}$-fat pairwise-disjoint regions with diameter at least 1 which can intersect $S$.

For example, if $r_2 = r_1=1$, then the regions are circles, and there are at most 6 circles at least as large as $S$ which can overlap $S$ without overlapping each other.

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 simplicity, let $r_1=1$, and let $2$ be the side length of a square $S$.

I want to know the maximum number of $\frac{r_2}{r_1}$-fat pairwise-disjoint regions with diameter at least 1 which can intersect $S$.

For example, if $r_2 = r_1=1$, then the regions are circles, and there are at most 6 circles with diameter at least 1 which can overlap $S$ without overlapping each other. In the figure below, we have depicted a square and 7 circles, but the circles above and below the square cannot overlap the square without also overlapping the circle in the center of the square.

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 thickness of $R$ is given by $\frac{r_2}{r_1}$, and we say that $R$ is $\beta$-thick, for $\beta = \frac{r_2}{r_1}$.

For simplicity, let $r_1=1$, and let $2$ be the side length of a square $S$.

I want to know the maximum number of $\frac{r_2}{r_1}$-thick pairwise-disjoint regions which can intersect $S$.

For example, if $r_2 = r_1=1$, then the regions are circles, and there are at most 6 circles at least as large as $S$ which can overlap $S$ without overlapping each other.

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 simplicity, let $r_1=1$, and let $2$ be the side length of a square $S$.

I want to know the maximum number of $\frac{r_2}{r_1}$-fat pairwise-disjoint regions with diameter at least 1 which can intersect $S$.

For example, if $r_2 = r_1=1$, then the regions are circles, and there are at most 6 circles at least as large as $S$ which can overlap $S$ without overlapping each other.