Extremal Problems Of L_p-Intersection Bodies And Geominimal Surface Area

This article builds the Lp-Brunn-Minkowski theory,studies the nature of convex body and star body in Euclidean space.We use basis knowledge and approaches of Lp—Brunn-Minkowski theory to make a further study of following aspects.In the first aspect,we research the properties of i-th Lp-dual affine siurface area.the other parts,we made a promotion to generalized intersection of Euclidean and research the extremal values of surface area.In the aspects of the i-th Lp-dual affine surface area,we according to the definition of Lp-mixed volume,Lp-dual mixed volume,Lp-affine surface area and Lp-dual affine surface area,combining with the definition and properties of i-th L,-affiine surfcace area by Ma Tongi,.we introduce the new concept of i-th Lp-dual affine surface area.Then combining with the Minkowski radial combination,Brunn-Minkowski-Fiery theory and Blaschke-Santalo inequality about i,-th Lp-dual affine surface area.In 2005,Ludwig and Haberl extend classical intersection and defined L,-intersection bodies.Further,Wang int,roduced concept of general Lp-intersection bodies and gen-eralized intersection bodies.Then,Ma Tongyi studied the generalized intersection aiind get,Some important,lprcoperties and ineciuality,combibining:with this(definition and achievement,we introduced concept of generalized intersection bodies with pa-rameter and get,some properties.On this basis,we also established inequality of the extremal values of the volume and some Brunn-Minkowski type inequality of radial combinations and Lp-harmonic Blaschke combinations.