Research On Some Geometric Inequalities In The Lp-space

This pa per belongs to the convex geometric analys is theory, which devoted to s tudying Lp-Bl aschke-Minkowski homomorphis ms, Lp-dual mixed affine surface area,Lp-dua l mixed geominimal surfa ce area for multiple star bodi es, Lp- inter section, quasi Lp-intersection. The main technologies used i n this paper are basic concepts, basic methods and integral tr ansforms in the Brunn-Minkowski t heory and its dua l theory.The main results can be stated as follows:1. We r esearc hered two monotonicity inequalities a nd Shephard type inequalities for the affine surfac e of Lp-Blas chke-Minkowski homomorphisms.2. Combing with the Lp-dual affine surfa ce, we defined the concept of Lp-dual mixed affine surface ar ea and e stablishe d some ine quali tie s.3. Based on t he Lp-mi xed geominimal surfac e are a for multiple convex bodies, we defined the concept of Lp-dual mi xed geominimal surface area for mul tiple star bodies and established some inequalit ies rela ted to this concept.4. Based on the conc ept of Lp-i ntersection body, we intr oduced the concept of gene ral Lp-inter section and re searchered its properties. Furthermore, we gave the extremum of volume and the Brunn-Minkowski type inequa lity relat ed to t his conce pt.5. Based on the notions of quasi Lp-intersection bodi es, we give star dual and Dres her-type i nequaliti es of quas i Lp-intersection bodie s.