A short elementary proof of reversed Brunn–Minkowski inequality for coconvex bodies
François Fillastre1
1 Université de Cergy-Pontoise UMR CNRS 8088 95000 Cergy-Pontoise (France)
Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 93-96.

The theory of coconvex bodies was formalized by A. Khovanskiĭ and V. Timorin in [4]. It has fascinating relations with the classical theory of convex bodies, as well as applications to Lorentzian geometry. In a recent preprint [5], R. Schneider proved a result that implies a reversed Brunn–Minkowski inequality for coconvex bodies, with description of equality case. In this note we show that this latter result is an immediate consequence of a more general result, namely that the volume of coconvex bodies is strictly convex. This result itself follows from a classical elementary result about the concavity of the volume of convex bodies inscribed in the same cylinder.

DOI: 10.5802/tsg.356
Keywords: coconvex sets, covolume, Brunn–Minkowski
François Fillastre 1

1 Université de Cergy-Pontoise UMR CNRS 8088 95000 Cergy-Pontoise (France) 
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François Fillastre. A short elementary proof of reversed Brunn–Minkowski inequality for coconvex bodies. Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 93-96. doi : 10.5802/tsg.356. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.356/

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[2] Francesco Bonsante; François Fillastre The equivariant Minkowski problem in Minkowski space, Ann. Inst. Fourier, Volume 67 (2017) no. 3, pp. 1035-1113 | DOI | MR | Zbl

[3] François Fillastre Fuchsian convex bodies: basics of Brunn–Minkowski theory, Geom. Funct. Anal., Volume 23 (2013) no. 1, pp. 295-333 | DOI | MR | Zbl

[4] Askold Khovanskiĭ; Vladlen Timorin On the theory of coconvex bodies, Discrete Comput. Geom., Volume 52 (2014) no. 4, pp. 806-823 | DOI | MR | Zbl

[5] Rolf Schneider A Brunn–Minkowski theory for coconvex sets of finite volume, Adv. Math., Volume 332 (2018), pp. 199-234 | DOI | MR | Zbl

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