Open Access
2016 The Dual Log-Brunn-Minkowski Inequalities
Wei Wang, Lijuan Liu
Taiwanese J. Math. 20(4): 909-919 (2016). DOI: 10.11650/tjm.20.2016.6323
Abstract

In this article, we establish the dual log-Brunn-Minkowski inequality and the dual log-Minkowski inequality. Moreover, the equivalence between the dual log-Brunn-Minkowski inequality and the dual log-Minkowski inequality is demonstrated.

References

1.

K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), no. 3-4, 1974–1997.  10.1016/j.aim.2012.07.015 MR2964630 1258.52005 K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang, The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012), no. 3-4, 1974–1997.  10.1016/j.aim.2012.07.015 MR2964630 1258.52005

2.

W. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24.  MR141003 0188.27303 10.7146/math.scand.a-10510 W. J. Firey, $p$-means of convex bodies, Math. Scand. 10 (1962), 17–24.  MR141003 0188.27303 10.7146/math.scand.a-10510

3.

R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405.  10.1090/s0273-0979-02-00941-2 MR1898210 1019.26008 R. J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 3, 355–405.  10.1090/s0273-0979-02-00941-2 MR1898210 1019.26008

4.

––––, Geometric Tomography, Second edition, Encyclopedia of Mathematics and its Applications, 58, Cambridge University Press, Cambridge, 2006.  10.1017/cbo9781107341029 ––––, Geometric Tomography, Second edition, Encyclopedia of Mathematics and its Applications, 58, Cambridge University Press, Cambridge, 2006.  10.1017/cbo9781107341029

5.

P. M. Gruber, Convex and Discrete Geometry, Grundlehren der Mathematischen Wissenschaften, 336, Springer, Berlin, 2007.  10.1007/978-3-540-71133-9 1139.52001 P. M. Gruber, Convex and Discrete Geometry, Grundlehren der Mathematischen Wissenschaften, 336, Springer, Berlin, 2007.  10.1007/978-3-540-71133-9 1139.52001

6.

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Reprint of the 1952 edition, Cambridge mathematical Library, Cambridge University Press, Cambridge, 1988. G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Reprint of the 1952 edition, Cambridge mathematical Library, Cambridge University Press, Cambridge, 1988.

7.

E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531–538.  10.2140/pjm.1975.58.531 MR380631 0273.52007 euclid.pjm/1102905685 E. Lutwak, Dual mixed volumes, Pacific J. Math. 58 (1975), no. 2, 531–538.  10.2140/pjm.1975.58.531 MR380631 0273.52007 euclid.pjm/1102905685

8.

––––, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150. ––––, The Brunn-Minkowski-Firey theory I: Mixed volumes and the Minkowski problem, J. Differential Geom. 38 (1993), no. 1, 131–150.

9.

R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993.  10.1017/cbo9780511526282 R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Encyclopedia of Mathematics and its Applications, 44, Cambridge University Press, Cambridge, 1993.  10.1017/cbo9780511526282
Copyright © 2016 The Mathematical Society of the Republic of China
Wei Wang and Lijuan Liu "The Dual Log-Brunn-Minkowski Inequalities," Taiwanese Journal of Mathematics 20(4), 909-919, (2016). https://doi.org/10.11650/tjm.20.2016.6323
Published: 2016
Vol.20 • No. 4 • 2016
Back to Top