Proof of the Brunn–Minkowski Theorem by Elementary Methods - Malyshev, F. M. (Journal of mathematical sciences (New York, N.Y.), 2023)
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 ℝ n , n ≥ 2: V n ((1 − t ) P 0 + tP 1 ) ≥ 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 translation; 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 ⊂ ℝ n , we construct a continuous (in the Hausdorff metric) one-parameter family of convex polyhedra P 1 ( s ) ⊂ ℝ n , s ∈ [0 , 1], P 1 (0) = P 1 , for which the function w ( s ) = V n ((1 − t ) P 0 + tP 1 ( s )) strictly monotonically decreases, and P 1 (1) is obtained from P 0 by a parallel translation. If P 1 is not obtained from P 0 by a parallel translation, then, using elementary geometric constructions, we establish the existence of a polyhedron P 1 ′ for which V n ((1 − t ) P 0 + tP 1 ) > V n ((1 − t ) P 0 + t P 1 ′ ).