Inequalities And Extremum Problem For The Convex Body In The L_p-space

Our main works are to research the theories of convex bodies, some inequalitiesand extremum properties of some geometry bodies by applying the basic notions,basic theories and integral transforms of the L_p Brunn-Minkowski theory. Espe-cially, the application of the Busemann-Petty problem. This article belong to thefield, which is a high-speed developing geometry branch on the decade of late, of theL_p Brunn-Minkowski theory. The main results are as follows.（1）Lutwak introduced the notion of the i-th width-integral of convex bodies. Wegive the concept of the mixed width-integral of convex bodies and get some proper-ties. Besides, we establish its Circular inequality and Brunn-Minkowski inequality.（2）We introduce the concept of p mixed chord-integrals of star bodies and getsome properties, also establish its circular inequality and Brunn-Minkowski inequal-ity.（3）We define the notions of new geometric body L_p,iK and L_p mixed harmonicBlaschke add. We establish the Brunn-Minkowski inequalities for the quermassin-tegrals and dual quermassintegrals of Lp mixed centroid body Γp,iKand its polarbody associated with Lp mixed harmonic Blaschke add, and prove the monotonic-ity of operator Γ_p,iand Γ_p,i.（4）We will combine to promote the concepts of ith curvature function f_i（K,·）and L_p curvature function f_p（K,·） for a convex body K inRn, we introduce theconcept of L_p mixed curvature functionf_p,i（K,·） for a convex body K inRn. Andon this basis, we will consider the analog question of the Busemann-Petty problem,whether Γ_p,iK Γ-（p,i）L implies W_i（K）≤W_i（L）, is investigated.