Proof of the Brunn–Minkowski theorem by elementary methods

In this paper, we propose new proofs of the classical Brunn—Minkowski theorem on the volume of the sum of convex polyhedra $P_0$ and $P_1$ of the same $n$-dimensional volume in Euclidean space $\mathbb{R}^n$, $n\ge2$: $V_n((1-t)P_0+tP_1) \ge V_n(P_0) = V_n(P_1)$, $0<t<1$, where the equality holds only if $P_1$ is obtained from $P_0$ by a parallel transfer; in other cases, the strict inequality holds. Proofs are based on the sequential partition of the polyhedron $P_0$ into simplexes by hyperplanes. For dimensions $n=2$ and $n=3$, in the case where $P_0$ is a simplex (a triangle for $n=2$), for an arbitrary convex polyhedron $P_1 \subset \mathbb{R}^n$, we construct a continuous (in the Hausdorff metric) one-parameter family of convex polyhedra $P_1(s) \subset \mathbb{R}^n$, $s \in [0,1]$, $P_1(0)=P_1$, for which the function $w(s)=V_n\big((1-t) P_0 + tP_1 (s)\big)$ strictly monotonically decreases, and $P_1(1)$ is obtained from $P_0$ by a parallel transfer. If $P_1$ is not obtained from $P_0$ by a parallel transfer, then, using elementary geometric constructions, we establish the existence of a polyhedron $P_1'$ for which $V_n\big((1-t) P_0 + tP_1\big)> V_n \big((1-t ) P_0 + tP'_1\big)$.