The Orlicz Brunn-minkowski Inequality For Harmonic Quermassintegrals

The classical Brunn-Minkowski theory is one of the cornerstones of the convex geometry,and has been generalized as theL_p Brunn-Minkowski theory in recent decades.Very recently,this theory has been generalized as the Orlicz Brunn-Minkowski by Lutwak,Yang,Zhang and Gardner,Hug,Weil et al.This thesis belongs to the Orlicz Brunn-Minkowski.In this thesis,by calculating the first Orlicz variations of harmonic quermassintegrals,we introduce the notion of Orlicz mixed harmonic quermassintegrals,and prove an Orlicz Minkowski type inequality for these new geometric quantities.Furthermore,we establish an Orlicz Brunn-Minkowski type inequality for harmonic quermassintegrals.Our new Orlicz Minkowski type inequality and Orlicz Brunn-Minkowski type inequality generalize theL_p Minkowski type inequality andL_p Brunn-Minkowski inequality for harmonic quermassintegrals.In particular,if p(28)1,then our results reduce to the original of Hadwiger.Finally,a multiple Orlicz Brunn-Minkowski type inequality for harmonic quermassintegrals is established.