On the Brunn and Brunn–Minkowski theorems

The case of equality in the Brunn-Minkowski theorem presents the greatest difficulty for the reason that so far the proofs have been carried out not for three parallel sections of convex bodies, as in Brunn's, but for Minkowski sums. The cuts used by Brunn are sufficient for an elementary (accessible to schoolchildren) proof of the theorem. Scribe the case of equality to Minkowski. Keywords: convex bodies, polyhedra, Brunn-Minkowski inequality.