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General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem I

  • Richard J. Gardner
  • Daniel Hug
  • Wolfgang Weil
  • Sudan Xing
  • Deping YeEmail author
Article
  • 75 Downloads

Abstract

The general volume of a star body, a notion that includes the usual volume, the qth 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 (pq)-dual curvature measure introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combination. 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 (pq)-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 (pq)-mixed volume.

Mathematics Subject Classification

Primary: 52A20 52A30 Secondary: 52A39 and 52A40 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Richard J. Gardner
    • 1
  • Daniel Hug
    • 2
  • Wolfgang Weil
    • 2
  • Sudan Xing
    • 3
  • Deping Ye
    • 3
    Email author
  1. 1.Department of MathematicsWestern Washington UniversityBellinghamUSA
  2. 2.Department of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Department of Mathematics and StatisticsMemorial University of NewfoundlandNLCanada

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