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?

Intersecting balls
Intersecting balls

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.

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