Fix two positive numbers and . The volume of the intersection of balls of radii and is a function of the distance between their centers. Is it a decreasing function?
Wolfram MathWorld gives a formula for this volume in the three-dimensional case
(Here and in the sequel it is assumed that .) From this formula it is not at all obvious that is decreasing. And this is just for three dimensions.
Let’s begin with the one-dimensional case, then. Consider the intervals and where . A point belongs to their intersection if and only if
Hence, the length of the intersection is . Even this simple formula does not quite make it evident that the function is decreasing for . If this is clear enough, but the case .
Fortunately, Fubini’s theorem reduces the general case to the one-dimensional one. Take any line parallel to the line connecting the centers. The intersection of each ball with is a line segment whose length is independent of . The distance between the midpoints of these segments is ; thus, the length of the intersection is a decreasing function of .
Gromov’s note Monotonicity of the volume of intersection of balls (1987) deals with the more general case of balls in : if two sets of balls an satisfy for all , then the volume of does not exceed the volume of .
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.