General volumes in the Orlicz-Brunn-Minkowski theory and related Minkowski problems

Xing, Sudan (2019) General volumes in the Orlicz-Brunn-Minkowski theory and related Minkowski problems. Doctoral (PhD) thesis, Memorial University of Newfoundland.

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Abstract

The Minkowski problem is one of the core problems in convex geometry, which aims to characterize the surface area measures of convex bodies in Rn. Various extensions and their dual have been introduced in recent years, among which the newly proposed dual Minkowski problem for the qth dual curvature measure is one of the most important. This thesis deals with the general (dual) volumes, the general dual Orlicz curvature measure e CG,ψ, and related Minkowski type problems. A typical problem we investigate in this thesis is the following general dual Orlicz- Minkowski problem: under what conditions on a given measure μ defined on the unit sphere, a two-variable function G(·, ·) and a one-variable function ψ(·), does there exist a convex body K such that μ equals to the general dual Orlicz curvature measure of K up to a constant τ , i.e., μ = τ e CG,ψ(K, ·)? In particular, we will study the existence, continuity, and uniqueness of the solutions to the above general dual Orlicz- Minkowski problem. These will be done in Chapters 3-5, where Chapter 3 deals with the special case of e CG,ψ obtained from Vφ, Chapter 4 studies the case e CG,ψ with G(t, ·) decreasing on t, and Chapter 5 concentrates on the case e CG,ψ with G(t, ·) increasing on t. Techniques used in Chapters 3 and 4 are the Blaschke selection theorem and the method of Lagrange multipliers, whereas in Chapter 5 we use the approximation arguments from discrete measures to general measures. In Chapter 6, we investigate the “polar” of the general dual Orlicz-Minkowski problem. This type of problem is a typical extension of many fundamental geometric invariants, such as the Lp/Orlicz geominimal surface areas and the Lp/Orlicz-Petty bodies. The existence, continuity, and uniqueness of the solutions to the general dual-polar Orlicz-Minkowski problem are provided in Chapter 6. Our techniques also follow the approximation arguments from discrete measures to general measures.

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