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


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Article Info

DATE

2019-02

AUTHORS

Richard J. Gardner, Daniel Hug, Wolfgang Weil, Sudan Xing, Deping Ye

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 (p, q)-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 (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. More... »

PAGES

12

References to SciGraph publications

  • 2016-06. Geometric measures in the dual Brunn–Minkowski theory and their associated Minkowski problems in ACTA MATHEMATICA
  • 2018-12. The Dual Orlicz–Minkowski Problem in THE JOURNAL OF GEOMETRIC ANALYSIS
  • 2018-02. The p-capacitary Orlicz–Hadamard variational formula and Orlicz–Minkowski problems in CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS
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    http://scigraph.springernature.com/pub.10.1007/s00526-018-1449-0

    DOI

    http://dx.doi.org/10.1007/s00526-018-1449-0

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1110344426


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