The volume of the intersection of balls

Fix two positive numbers ${R}$ and ${r}$. The volume of the intersection of balls of radii ${R}$ and ${r}$ is a function of the distance ${d}$ between their centers. Is it a decreasing function?

Wolfram MathWorld gives a formula for this volume in the three-dimensional case

$\displaystyle V(d) = \frac{\pi}{12 d} (R+r-d)^2(d^2+2rd-3r^2+2Rd+6Rr-3R^2)$

(Here and in the sequel it is assumed that ${d\le R+r}$.) From this formula it is not at all obvious that ${V(d)}$ is decreasing. And this is just for three dimensions.

Let’s begin with the one-dimensional case, then. Consider the intervals ${[-R,R]}$ and ${[d-r,d+r]}$ where ${d\le R+r}$. A point ${x}$ belongs to their intersection if and only if

$\displaystyle \max(-R,d-r)\le x \le \min(R,d+R)$

Hence, the length of the intersection is ${L(d)=\min(R,d+r)-\max(-R,d-r)}$. Even this simple formula does not quite make it evident that the function is decreasing for ${d>0}$. If ${d+r>R}$ this is clear enough, but the case ${d+r -d-r>-r}$.

Fortunately, Fubini’s theorem reduces the general case to the one-dimensional one. Take any line ${L}$ parallel to the line connecting the centers. The intersection of each ball with ${L}$ is a line segment whose length is independent of ${d}$. The distance between the midpoints of these segments is ${d}$; thus, the length of the intersection is a decreasing function of ${d}$.

Gromov’s note Monotonicity of the volume of intersection of balls (1987) deals with the more general case of ${k\le n+1}$ balls in ${\mathbb R^n}$: if two sets of balls ${B(x_i,r_i)}$ an ${B(x_i',r_i)}$ satisfy ${\|x_i'-x_j'\|\le \|x_i-x_j\|}$ for all ${i,j}$, then the volume of ${\bigcap B(x_i,r_i)}$ does not exceed the volume of ${\bigcap B(x_i',r_i)}$.

The paper ends with

Question: Who is the author of [this result]? My guess is this was known to Archimedes. Undoubtedly the theorem can be located […] somewhere in the 17th century.

Meanwhile, the corresponding problem for the volume of the union remains open.

2 thoughts on “The volume of the intersection of balls”

1. Anonymous says:

i think you missed the assumption k \leq n+1 in Gromov’s theorem…

1. L says:

Thanks, fixed.