The log-Brunn-Minkowski inequality in ℝ³

Yang, Yunlong; Zhang, Deyan

doi:10.1090/proc/14366

The log-Brunn-Minkowski inequality in $\mathbb {R}^3$
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by Yunlong Yang and Deyan Zhang PDF

Proc. Amer. Math. Soc. 147 (2019), 4465-4475 Request permission

Abstract:

Böröczky, Lutwak, Yang, and Zhang have recently proved the log-Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane which is stronger than the classical Brunn-Minkowski inequality. This paper establishes the log-Brunn-Minkowski, log-Minkowski, $L_p$-Minkowski, and $L_p$-Brunn-Minkowski inequalities for two classes of convex bodies in $\mathbb {R}^3$.

References

Károly J. Böröczky and Martin Henk, Cone-volume measure of general centered convex bodies, Adv. Math. 286 (2016), 703–721. MR 3415694, DOI 10.1016/j.aim.2015.09.021
Térence Bayen and Didier Henrion, Semidefinite programming for optimizing convex bodies under width constraints, Optim. Methods Softw. 27 (2012), no. 6, 1073–1099. MR 2955296, DOI 10.1080/10556788.2010.547580
W. Blaschke, Einige Bemerkungen über Kurven und Flächen von konstanter Breite, Ber. Verh. sächs. Akad. Leipzig 67 (1915), 290–297.
Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), no. 3-4, 1974–1997. MR 2964630, DOI 10.1016/j.aim.2012.07.015
Károly J. Böröczky, Erwin Lutwak, Deane Yang, and Gaoyong Zhang, The logarithmic Minkowski problem, J. Amer. Math. Soc. 26 (2013), no. 3, 831–852. MR 3037788, DOI 10.1090/S0894-0347-2012-00741-3
G. D. Chakerian, Sets of constant width, Pacific J. Math. 19 (1966), 13–21. MR 205152
Andrea Colesanti, Galyna V. Livshyts, and Arnaud Marsiglietti, On the stability of Brunn-Minkowski type inequalities, J. Funct. Anal. 273 (2017), no. 3, 1120–1139. MR 3653949, DOI 10.1016/j.jfa.2017.04.008
A. Colesanti and G. V. Livshyts, A note on the quantitative local version of the log-Brunn-Minkowski inequality, submitted, arXiv:1710.10708, 2017.
Wm. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24. MR 141003, DOI 10.7146/math.scand.a-10510
Michael E. Gage, Evolving plane curves by curvature in relative geometries, Duke Math. J. 72 (1993), no. 2, 441–466. MR 1248680, DOI 10.1215/S0012-7094-93-07216-X
Michael E. Gage and Yi Li, Evolving plane curves by curvature in relative geometries. II, Duke Math. J. 75 (1994), no. 1, 79–98. MR 1284816, DOI 10.1215/S0012-7094-94-07503-0
R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405. MR 1898210, DOI 10.1090/S0273-0979-02-00941-2
Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
Martin Henk and Eva Linke, Cone-volume measures of polytopes, Adv. Math. 253 (2014), 50–62. MR 3148545, DOI 10.1016/j.aim.2013.11.015
Binwu He, Gangsong Leng, and Kanghai Li, Projection problems for symmetric polytopes, Adv. Math. 207 (2006), no. 1, 73–90. MR 2264066, DOI 10.1016/j.aim.2005.11.006
Martin Henk, Achill Schürmann, and Jörg M. Wills, Ehrhart polynomials and successive minima, Mathematika 52 (2005), no. 1-2, 1–16 (2006). MR 2261838, DOI 10.1112/S0025579300000292
A. V. Kolesnikov and E. Milman, Local $L_p$-Brunn-Minkowski inequalities for $p<1$, arXiv:1711.01089, 2017.
Erwin Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. MR 1231704
Erwin Lutwak, The Brunn-Minkowski-Firey theory. II. Affine and geominimal surface areas, Adv. Math. 118 (1996), no. 2, 244–294. MR 1378681, DOI 10.1006/aima.1996.0022
Lei Ma, A new proof of the log-Brunn-Minkowski inequality, Geom. Dedicata 177 (2015), 75–82. MR 3370024, DOI 10.1007/s10711-014-9979-x
Shengliang Pan, Deyan Zhang, and Yunlong Yang, Two measures of asymmetry for revolutionary constant width bodies in $\Bbb {R}^3$, J. Math. Anal. Appl. 433 (2016), no. 2, 852–861. MR 3398740, DOI 10.1016/j.jmaa.2015.08.016
L. Rotem, A letter: The log-Brunn-Minkowski inequality for complex bodies, arXiv:1412.5321, 2014.
Luis A. Santaló, Integral geometry and geometric probability, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004. With a foreword by Mark Kac. MR 2162874, DOI 10.1017/CBO9780511617331
Christos Saroglou, Remarks on the conjectured log-Brunn-Minkowski inequality, Geom. Dedicata 177 (2015), 353–365. MR 3370038, DOI 10.1007/s10711-014-9993-z
Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
Alina Stancu, The discrete planar $L_0$-Minkowski problem, Adv. Math. 167 (2002), no. 1, 160–174. MR 1901250, DOI 10.1006/aima.2001.2040
Alina Stancu, On the number of solutions to the discrete two-dimensional $L_0$-Minkowski problem, Adv. Math. 180 (2003), no. 1, 290–323. MR 2019226, DOI 10.1016/S0001-8708(03)00005-7
Alina Stancu, The logarithmic Minkowski inequality for non-symmetric convex bodies, Adv. in Appl. Math. 73 (2016), 43–58. MR 3433500, DOI 10.1016/j.aam.2015.09.015
A. C. Thompson, Minkowski geometry, Encyclopedia of Mathematics and its Applications, vol. 63, Cambridge University Press, Cambridge, 1996. MR 1406315, DOI 10.1017/CBO9781107325845
Dongmeng Xi and Gangsong Leng, Dar’s conjecture and the log-Brunn-Minkowski inequality, J. Differential Geom. 103 (2016), no. 1, 145–189. MR 3488132
Ge Xiong, Extremum problems for the cone volume functional of convex polytopes, Adv. Math. 225 (2010), no. 6, 3214–3228. MR 2729006, DOI 10.1016/j.aim.2010.05.016
Guangxian Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math. 262 (2014), 909–931. MR 3228445, DOI 10.1016/j.aim.2014.06.004