General volumes in the Orlicz–Brunn–Minkowski theory and a related Minkowski problem II
By Richard J. Gardner, Daniel Hug, Sudan Xing, Deping Ye. Calculus of Variations and Partial Differential Equations, 2018
88 citations recorded by the source corpus.
For convex bodies K in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} containing the origin in their interiors, the general dual volume and the general dual Orlicz curvature measure C~G,ψ(K,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{C}_{G, \psi }(K, \cdot )$$\end{document} were recently introduced for certain classes of functions G and ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}. We extend these concepts to more general functions G and to compact convex sets K containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure are provided which are required to study a Minkowski-type problem for the dual Orlicz curvature measure. The Minkowski problem asks to characterize Borel measures μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} on the unit sphere for which there is a convex body K in Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} containing the origin such that μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} equals C~G,ψ(K,·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widetilde{C}_{G, \psi }(K, \cdot )$$\end{document}, up to a constant. A major step in the analysis concerns discrete measures μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document}, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. Under mild conditions on G and ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}, solutions are obtained for general measures by an approximation argument. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} is discrete or even.
Subfields: Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/c0a1df26f378c54f037656750cc10a33b77b130d
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Applied Mathematics
Citations in the corpus
- The Minkowski problem in Gaussian probability space (2020, background): “The author would like to point out that a differently normalized version of the Gaussian Minkowski problem is a special case of the general dual Orlicz-Minkowski problem considered in [23, 24].”
- A flow method for the dual Orlicz–Minkowski problem (2020, methodology): “For convenience, we here have used a slightly different notation in the definition of C̃φ,G(K, ·) from that of [21, 23].”