Dual mixed Orlicz-Brunn-Minkowski inequality and dual Orlicz mixed quermassintegrals
In [Adv. Math. 223, No. 1, 220--242 (2010; Zbl 1437.52006)], \textit{E. Lutwak} et al. introduced the Orlicz-Brunn-Minkowski theory as a far reaching extension of the classical Brunn-Minkowski theory. Since then it has been investigated in numerous papers and it has been proved a very powerful theory within modern convex geometry (see, e.g., [\textit{R. J. Gardner} et al., J. Differ. Geom. 97, No. 3, 427--476 (2014; Zbl 1303.52002)]). In analogy to the interplay between classical Brunn-Minkowski theory and the dual Brunn-Minkowski theory introduced by \textit{E. Lutwak} [Pac. J. Math. 58, 531--538 (1975; Zbl 0273.52007)], in recent years a dual Orlicz-Brunn-Minkowski theory was developed by various authors, see, e.g., [\textit{R. J. Gardner} et al., J. Math. Anal. Appl. 430, No. 2, 810--829 (2015; Zbl 1320.52008); \textit{C. Lange}, in: Proceedings of the 23rd international conference on formal power series and algebraic combinatorics, FPSAC'11. Nancy: The Association. Discrete Mathematics \& Theoretical Computer Science (DMTCS). 611--622 (2011; Zbl 1355.52008); \textit{B. Zhu} et al., Adv. Math. 264, 700--725 (2014; Zbl 1307.52004)]. In the paper under review, the authors establish within the dual Orlicz-Brunn-Minkowski theory first a dual mixed Orlicz-Brunn-Minkowski inequality, i.e., a Brunn-Minkowski type inequality for the dual quermassintegral of the Orlicz radial sum of two star bodies, and second, a dual Orlicz-Minkowski inequality for the so called dual Orlicz mixed quermassintegrals which are introduced by the authors.
dual mixed Orlicz-Brunn-Minkowski inequality
scientific article; zbMATH DE number 6755362
scientific article