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.
@article{TSG_2016-2017__34__93_0,
author = {Fillastre, Fran\c{c}ois},
title = {A short elementary proof of reversed {Brunn{\textendash}Minkowski} inequality for coconvex bodies},
journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
pages = {93--96},
publisher = {Institut Fourier},
address = {Grenoble},
volume = {34},
year = {2016-2017},
doi = {10.5802/tsg.356},
language = {en},
url = {http://www.numdam.org/articles/10.5802/tsg.356/}
}
TY - JOUR AU - Fillastre, François TI - A short elementary proof of reversed Brunn–Minkowski inequality for coconvex bodies JO - Séminaire de théorie spectrale et géométrie PY - 2016-2017 SP - 93 EP - 96 VL - 34 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/tsg.356/ DO - 10.5802/tsg.356 LA - en ID - TSG_2016-2017__34__93_0 ER -
%0 Journal Article %A Fillastre, François %T A short elementary proof of reversed Brunn–Minkowski inequality for coconvex bodies %J Séminaire de théorie spectrale et géométrie %D 2016-2017 %P 93-96 %V 34 %I Institut Fourier %C Grenoble %U http://www.numdam.org/articles/10.5802/tsg.356/ %R 10.5802/tsg.356 %G en %F TSG_2016-2017__34__93_0
Fillastre, François. A short elementary proof of reversed Brunn–Minkowski inequality for coconvex bodies. Séminaire de théorie spectrale et géométrie, Tome 34 (2016-2017), pp. 93-96. doi : 10.5802/tsg.356. http://www.numdam.org/articles/10.5802/tsg.356/
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