The notion of the mixed brightness was first introduced by Lutwak. Recently, Abardia and Bernig presented complex projection bodies. Based on this notion, we define the mixed complex brightness integrals and establish related Aleksandrov–Fenchel inequality, cyclic inequality and monotonicity inequality, respectively.
This is a preview of subscription content, log in to check access.
Buy single article
Instant unlimited access to the full article PDF.
Price includes VAT for USA
Abardia, J.: Difference bodies in complex vector spaces. J. Funct. Anal. 263, 3588–3603 (2012)
Abardia, J.: Minkowski valuations in a 2-dimensional complex vector space. Int. Math. Res. Not. 2015, 1247–1262 (2015)
Abardia, J., Bernig, A.: Projection bodies in complex vector spaces. Adv. Math. 263, 3588–3603 (2012)
Bernig, A., Fu, J.H.G., Solanes, G.: Integral geometry of complex space forms. Geom. Funct. Anal. 24, 403–492 (2014)
Federer, H.: Geometric Measure Theory. Springer, New York (1969)
Gardner, R.J.: Geometric Tomography, 2nd edn. Cambridge University Press, Cambridge (2006)
Haberl, C., Schuster, F.: General \(L_p\) affine isoperimetric inequalities. J. Differ. Geom. 83, 1–26 (2008)
Hardy, G.H., Littlewood, J.E., Pǒlya, G.: Inequalities. Cambridge University Press, Cambridge (1952)
Huang, Q.Z., He, B.W.: Volume inequalities for complex isotropic measures. Geom. Dedic. 177, 401–428 (2015)
Huang, Q.Z., He, B.W., Wang, G.T.: The Busemann theorem for complex \(p\)-convex bodies. Arch. Math. 99, 289–299 (2012)
Huang, Q.Z., Li, A.J., Wang, W.: The complex \(L_p\) Loomis-whitney inequality. Math. Inequal. Appl. 21, 369–383 (2018)
Koldobsky, A., König, H., Zymonopoulou, M.: The complex Busemann-Petty problem on sections of convex bodies. Adv. Math. 218, 352–367 (2008)
Koldobsky, A., Paouris, G., Zymonopoulou, M.: Complex intersection bodies. J. Lond. Math. Soc. 88, 538–562 (2013)
Koldobsky, A., Zymonopoulou, M.: Extremal sections of complex \(l_p\)-ball, \(0<p\le 2\). Stud. Math. 159, 185–194 (2003)
Ludwig, M.: Minkowski valuations. Trans. Am. Math. Soc. 357, 4191–4213 (2005)
Lutwak, E.: Mixed projection inequalities. Trans. Am. Math. Soc. 287, 91–106 (1985)
Li, N., Zhu, B.C.: Mixed brightness-integrals of convex bodies. J. Korean Math. Soc. 47, 935–945 (2010)
Liu, L.J., Wang, W., Huang, Q.Z.: On polars of mixed complex projection bodies. Bull. Korean Math. Soc. 52, 453–465 (2015)
Popoviciu, T.: On an inequality. Gaz. Mat. Fiz. Ser. 64, 451–461 (1959)
Rubin, B.: Comparison of volumes of convex bodies in real, complex, and quaternionic spaces. Adv. Math. 225, 1461–1498 (2010)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, 2nd edn. Cambridge University Press, Cambridge (2014)
Wu, D.H., Bu, Z.H., Ma, T.Y.: Two complex combinations and complex intersection bodies. Taiwan. J. Math. 18, 1459–1480 (2014)
Wang, W., He, R.G.: Inequalities for mixed complex projection bodies. Taiwan. J. Math. 17, 1887–1899 (2013)
Wang, W., He, R.G., Yuan, J.: Mixed complex intersection bodies. Math. Inequal. Appl. 18, 419–428 (2015)
Wang, W.D., Wan, X.Y.: Shephard type problems for general \(L_p\)-projection bodies. Taiwan. J. Math. 16, 1749–1762 (2012)
Wang, W.D., Feng, Y.B.: A general \(L_p\)-version of Petty’s affine projection inequality. Taiwan. J. Math. 17, 517–528 (2013)
Wang, W.D., Wang, J.Y.: Extremum of geometric functionals involving general \(L_p\)-projection bodies. J. Inequal. Appl. 2016, 1–16 (2016)
Yan, L., Wang, W.D.: General \(L_p\)-mixed-brightness integrals. J. Inequal. Appl. 2015, 1–11 (2015)
Zhao, C.J.: On mixed brightness-integrals. Rev. De La Unión Mat. Argent. 54, 27–34 (2013)
Zhao, C.J.: Volume differences of mixed complex projection bodies. Bull. Belg. Math. Soc. 21, 553–564 (2014)
Zhao, C.J.: On mixed complex intersection bodies. J. Nonlinear Sci. Appl. 11, 541–549 (2018)
Zymonopoulou, M.: The complex Busemann-Petty problem for arbitrary measures. Arch. Math. 91, 436–449 (2008)
Zymonopoulou, M.: The modified complex Busemann–Petty problem on sections of convex bodies. Positivity 13, 717–733 (2009)
Zhou, Y.P., Wang, W.D., Feng, Y.B.: The Brunn–Minkowski type inequalities for mixed brightness-integrals. Wuhan Univ. J. Nat. Sci. 19, 277–282 (2014)
The authors want to express earnest thankfulness for the referees who provided extremely precious and helpful comments and suggestions.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Chao Li: Research is supported in part by the Innovation Foundation of Graduate Student of China Three Gorges University (No.2019SSPY146).
Weidong Wang: Research is supported in part by the Natural Science Foundation of China (No.11371224).
Youjiang Lin: Research is supported in part by the funds of the Basic and Advanced Research Project of CQ CSTC (No.cstc2015jcyjA00009) and Scientific and Technological Research Program of Chongqing Municipal Education Commission (No.KJ1500628).
About this article
Cite this article
Li, C., Wang, W. & Lin, Y. Mixed complex brightness integrals. Positivity 24, 55–67 (2020) doi:10.1007/s11117-019-00665-5
- Complex projection body
- Mixed complex brightness integral
- Aleksandrov–Fenchel inequality
- Cyclic inequality
- Monotonicity inequality
Mathematics Subject Classification