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 Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 2020, Volume 182, Pages 70–94 (Mi into676)

Proof of the Brunn–Minkowski theorem by elementary methods

F. M. Malyshev

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: 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 $w(s)=V_n((1-t) P_0 + tP_1 (s))$ 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 $V_n((1-t) P_0 + tP_1)> V_n ((1-t ) P_0 + tP'_1)$.

Keywords: convex polytope, simplex, triangle, volumes, Brunn—Minkowski inequality

DOI: https://doi.org/10.36535/0233-6723-2020-182-70-94

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UDC: 514.172.4, 514.177.2
MSC: 52A20, 52A40

Citation: F. M. Malyshev, “Proof of the Brunn–Minkowski theorem by elementary methods”, Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 4, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 182, VINITI, Moscow, 2020, 70–94

Citation in format AMSBIB
\Bibitem{Mal20} \by F.~M.~Malyshev \paper Proof of the Brunn--Minkowski theorem by elementary methods \inbook Proceedings of the International Conference "Classical and Modern Geometry" Dedicated to the 100th Anniversary of the Birth of Professor Vyacheslav Timofeevich Bazylev. Moscow, April 22-25, 2019. Part 4 \serial Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz. \yr 2020 \vol 182 \pages 70--94 \publ VINITI \publaddr Moscow \mathnet{http://mi.mathnet.ru/into676} \crossref{https://doi.org/10.36535/0233-6723-2020-182-70-94}