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- Title
Proof of the Brunn–Minkowski Theorem by Elementary Methods.
- Authors
Malyshev, F. M.
- 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: Vn((1 − t)P0 + tP1) ≥ Vn(P0) = Vn(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) = Vn((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 ′ for which Vn ((1 − t)P0 + tP1) > Vn((1 − t)P0 + t P 1 ′ ).
- Subjects
GEOMETRICAL constructions; POLYHEDRA; HYPERPLANES; TRIANGLES
- Publication
Journal of Mathematical Sciences, 2023, Vol 277, Issue 5, p774
- ISSN
1072-3374
- Publication type
Article
- DOI
10.1007/s10958-023-06887-z