Geodesic
Proof of the Brunn--Minkowski Theorem by Brunn Cuts
Matematičeskie zametki, Tome 111 (2022) no. 1, pp. 80-92.

Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that for an exhaustive proof of the Brunn–Minkowski theorem on three parallel sections of a convex body, which states that if the areas of the extreme sections are equal, then the area of the middle section is strictly larger, it suffices to repeatedly apply Brunn's technique of cutting the body into two parts by a plane intersecting the three secant planes. If the body is not a cylinder, as is assumed in the theorem, then eliminating the case of equality can be explained to schoolchildren. The proposed elementary proof for arbitrary dimension refutes the common opinion that the case of equality in the theorem is special and most difficult to justify.
Keywords: convex body, polyhedron, Brunn–Minkowski inequality.
Mots-clés : simplex, volume 
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F. M. Malyshev. Proof of the Brunn--Minkowski Theorem by Brunn Cuts. Matematičeskie zametki, Tome 111 (2022) no. 1, pp. 80-92. http://geodesic.mathdoc.fr/item/MZM_2022_111_1_a7/

[1] H. Brunn, Uber Ovale und Eiflachen, Inag. Diss., Munchen, 1887

[2] H. Minkowski, Geometrie der Zahlen, B.G. Teubner, Leipzig und Berlin, 1910 | Zbl

[3] B. N. Delone, “Dokazatelstvo neravenstva Bruna–Minkovskogo”, UMN, 1936, no. 2, 39–46 | Zbl

[4] Yu. D. Burago, V. A. Zalgaller, Geometricheskie neravenstva, Nauka, L., 1980

[5] G. Federer, Geometricheskaya teoriya mery, Nauka, M., 1987

[6] V. V. Buldygin, A. B. Kharazishvili, Neravenstvo Brunna–Minkovskogo i ego prilozheniya, Naukova Dumka, Kiev, 1985

[7] R. J. Gardner, “The Brunn–Minkowski inequality”, Bull. Amer. Math. Soc. (N.S.), 39:3 (2002), 355–405 | DOI | Zbl

[8] G. Khadviger, Lektsii ob ob'eme, ploschadi poverkhnosti i izometrii, Nauka, M., 1966

[9] K. Leikhtveis, Vypuklye mnozhestva, Nauka, M., 1985 | MR | Zbl

[10] V. Blyashke, Krug i shar, Nauka, M., 1967 | MR | Zbl

[11] R. Schneider, “Convex Bodies: The Brunn–Minkowski theory”, Encyclopedia of Math. Appl., 151, Cambridge Univ. Press, Cambridge, 2013 | MR

[12] F. Barthe, “The Brunn–Minkowski theorem and related geometric and functional inequalities”, Proceedings of the International Congress of Mathematicians, European Math. Soc., 2006, 1529–1546 | Zbl

[13] K. Ball, “An Elementary introduction to monotone transportation”, Geometric Aspects of Functional Analysis, Lecture Notes in Math., 1850, Springer, Berlin, 2004, 41–52 | DOI | Zbl

[14] A. D. Aleksandrov, Vypuklye mnogogranniki, GITTL, M.-L., 1950

[15] R. J. Gardner, D. Hug, W. Weil, “The Orlicz–Brunn–Minkowski theory: A general framework, additions, and inequalities”, J. Differential. Geom., 2014, no. 3, 427–476 | MR | Zbl

[16] B. Berndtsson, “A Rrunn–Minkowski type inequalities for Fano manifolds and some uniqueness theorems in Kahler geometry”, Invent. Math., 200:1 (2015), 149–200 | DOI | Zbl

[17] B. S. Timergaliev, “Generalization of the Brunn–Vinkowski inequalitybin the Form of Hadwiger for Power Moments”, Lobachevskii J. Math., 37:6 (2016), 794–806 | DOI | Zbl

[18] F. M. Malyshev, “Dokazatelstva teoremy Brunna—Minkovskogo elementarnymi metodami”, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 182, VINITI RAN, M., 2020, 70–94 | DOI

[19] F. M. Malyshev, “Zavershenie dokazatelstva teoremy Brunna elementarnymi sredstvami”, Chebyshevskii sb., 22:2 (2021), 160–182 | DOI | Zbl

[20] E. G. Belousov, Vvedenie v vypuklyi analiz i tselochislennoe programmirovanie, Izd-vo Mosk. un-ta, M., 1977