Proof of the Brunn–Minkowski Theorem by Brunn Cuts

It is shown that for an exhaustive proof of the Brunn–Minkowski theorem on three parallel sections of a convex body, which states that if the areas of the extreme sections are equal, then the area of the middle section is strictly larger, it suffices to repeatedly apply Brunn's technique of cutting the body into two parts by a plane intersecting the three secant planes. If the body is not a cylinder, as is assumed in the theorem, then eliminating the case of equality can be explained to schoolchildren. The proposed elementary proof for arbitrary dimension refutes the common opinion that the case of equality in the theorem is special and most difficult to justify.