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General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I

Richard J. Gardner Western Washington University Daniel Hug Karlsruhe Institute of Technology Wolfgang Weil Karlsruhe Institute of Technology Sudan Xing Memorial University of Newfoundland Deping Ye Memorial University of Newfoundland

Analysis of PDEs Functional Analysis Geometric Analysis and Geometric Topology Convex and Discrete Geometry mathscidoc:1904.03004

Calc. Var. PDE., 58, 12, 2019
The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.
curvature measure, dual curvature measure, Minkowski problem, Orlicz addition, Orlicz-Brunn-Minkowski theory
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  • This paper provides a systematic study on the general dual Orlicz-Minkowski problems, which are arguably the most general Minkowski type problems and include all previously extensively studied Minkowski type problems as special cases (such as the $L_p$ Minkowski problem, the Orlicz-Minkowski problem, the dual Minkowski problem, the $L_p$ dual Minkowski problem, and the Aleksandrov problem). 
@inproceedings{richard2019general,
  title={General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I},
  author={Richard J. Gardner, Daniel Hug, Wolfgang Weil, Sudan Xing, and Deping Ye},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190418055750335035241},
  booktitle={Calc. Var. PDE.},
  volume={58},
  pages={12},
  year={2019},
}
Richard J. Gardner, Daniel Hug, Wolfgang Weil, Sudan Xing, and Deping Ye. General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I. 2019. Vol. 58. In Calc. Var. PDE.. pp.12. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190418055750335035241.
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