In this paper we revisit the anisotropic isoperimetric and the Brunn−Minkowski inequalities for convex sets. The best known constant C(n) = Cn7 depending on the space dimension n in both inequalities is due to Segal [A. Segal, Lect. Notes Math., Springer, Heidelberg 2050 (2012) 381–391]. We improve that constant to Cn6 for convex sets and to Cn5 for centrally symmetric convex sets. We also conjecture, that the best constant in both inequalities must be of the form Cn2, i.e., quadratic in n. The tools are the Brenier’s mapping from the theory of mass transportation combined with new sharp geometric-arithmetic mean and some algebraic inequalities plus a trace estimate by Figalli, Maggi and Pratelli.
Accepté le :
DOI : 10.1051/cocv/2017004
Mots-clés : Brunn–Minkowski inequality, wulff inequality, isoperimetric inequality, convex bodies
@article{COCV_2018__24_2_479_0,
author = {Harutyunyan, Davit},
title = {Quantitative anisotropic isoperimetric and {Brunn\ensuremath{-}Minkowski} inequalities for convex sets with improved defect estimates},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {479--494},
publisher = {EDP-Sciences},
volume = {24},
number = {2},
year = {2018},
doi = {10.1051/cocv/2017004},
zbl = {1406.52019},
mrnumber = {3816402},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2017004/}
}
TY - JOUR AU - Harutyunyan, Davit TI - Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 479 EP - 494 VL - 24 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2017004/ DO - 10.1051/cocv/2017004 LA - en ID - COCV_2018__24_2_479_0 ER -
%0 Journal Article %A Harutyunyan, Davit %T Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 479-494 %V 24 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2017004/ %R 10.1051/cocv/2017004 %G en %F COCV_2018__24_2_479_0
Harutyunyan, Davit. Quantitative anisotropic isoperimetric and Brunn−Minkowski inequalities for convex sets with improved defect estimates. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 2, pp. 479-494. doi : 10.1051/cocv/2017004. http://www.numdam.org/articles/10.1051/cocv/2017004/
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