Min ZhangStony Brook UniversityRen GuoOregon State UniveristyWei ZengSchool of Computing and Information Sciences, Florida International UniversityFeng LuoRutgers UniversityShing Tung YauHarvard UniversityXianfeng GuStony Brook Univerisity
Computational GeometryDifferential GeometryGeometric Modeling and ProcessingConvex and Discrete Geometry mathscidoc:1612.01001
Graphical Models/Geometric Modeling and Processing 2014, 76, (5), 321-339, 2014.9
Ricci ﬂow deformsthe Riemannian metric proportionallyto the curvature, such that the curvatureevolves accordingto a heat diffusion process and eventually becomes constant everywhere. Ricci ﬂow has demonstrated its great potential by solving various problems in many ﬁelds, which can be hardly handled by alternative methods so far. This work introduces the uniﬁed theoretic framework for discrete Surface Ricci Flow, including all the common schemes: Tangential Circle Packing, Thurston’s Circle Packing, Inversive Distance Circle Packing and Discrete Yamabe Flow. Furthermore, this work also introduces a novel schemes, Virtual Radius Circle Packing and the Mixed Type schemes, under the uniﬁed framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The uniﬁed frame work deepens our understanding to the the discrete surface Ricci ﬂow theory, and has inspired us to discover the new schemes, improved the ﬂexibility and robustness of the algorithms, greatly simpliﬁed the implementation and improved the efﬁciency. Experimental results show the uniﬁed surface Ricci ﬂow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems.
In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are estab-lished. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.
A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.
In this paper, we establish the $L_p$ Loomis-Whitney inequality for
even isotropic measures in terms of the support function of $L_p$
projection bodies with complete equality conditions. This
generalizes Ball's Loomis-Whitney inequality to the $L_p$ setting.
In addition, the sharp upper bound of the minimal $p$-mean width of
$L_p$ zonoids is obtained.