Brunn–Minkowski theorem

Remarks

For compact $A$ and $B$ the set $A + B$ is compact, hence measurable; for general measurable sets $A + B$ need not be measurable. One proof first checks boxes, where the inequality is the AM–GM inequality for the side ratios. It then passes to finite unions of disjoint boxes by induction on the number of boxes, cutting both sets by a coordinate hyperplane in matching proportions. Compact sets are then approximated from outside by such unions. Another proof applies the Prékopa–Leindler inequality to indicator functions. For compact convex sets of positive volume, equality holds if and only if $A$ and $B$ are homothetic. Example: for balls of radii $r$ and $s$, $A + B$ is a ball of radius $r + s$, and both sides equal $\omega_n^{1/n} (r + s)$, where $\omega_n$ is the volume of the unit ball.

Going deeper

For a student

The function $t \mapsto \lambda((1-t) A + t B)^{1/n}$ is concave on $[0, 1]$ for convex $A$ and $B$; this concavity form goes back to Brunn. Applied to $A$ and a small ball, the theorem gives the isoperimetric inequality. It is the starting point of the theory of mixed volumes, and it has a functional form, the Prékopa–Leindler inequality, used in probability and convex geometry.