Proof of the Brunn–Minkowski Theorem by Elementary Methods

Abstract

In this paper, we propose new proofs of the classical Brunn—Minkowski theorem on the volume of the sum of convex polyhedra P₀ and P₁ of the same n-dimensional volume in Euclidean space ℝⁿ, n ≥ 2: V_n((1 − t)P₀ + tP₁) ≥ V_n(P₀) = V_n(P₁), 0 < t < 1, where the equality holds only if P₁ is obtained from P₀ by a parallel translation; in other cases, the strict inequality holds. Proofs are based on the sequential partition of the polyhedron P₀ into simplexes by hyperplanes. For dimensions n = 2 and n = 3, in the case where P₀ is a simplex (a triangle for n = 2), for an arbitrary convex polyhedron P₁ ⊂ ℝⁿ, we construct a continuous (in the Hausdorff metric) one-parameter family of convex polyhedra P₁(s) ⊂ ℝⁿ, s ∈ [0, 1], P₁(0) = P₁, for which the function w(s) = V_n((1 − t)P₀ + tP₁(s)) strictly monotonically decreases, and P₁(1) is obtained from P₀ by a parallel translation. If P₁ is not obtained from P₀ by a parallel translation, then, using elementary geometric constructions, we establish the existence of a polyhedron ${P}_{1}{\prime}$ for which V_n ((1 − t)P₀ + tP₁) > V_n((1 − t)P₀ + t ${P}_{1}{\prime}$).