Metric Theoretical Research In Convex Bodies Geometry Of L_p-Space

The thesis is devoted to the study of metric inequalities and extremum problems in convex bodies geometry of L_p-space,and belongs to the domain,which is a high-speed developing geometry branch on the decade of late,of the L_p Brunn-Minkowski theory(or called Brunn-Minkowski-Firey theory).By applying the basic notions,basic theories and integral transforms of the L_p Brunn-Minkowski theory,we reseach the metric inequalities and extremum properties of some geometry bodies containing the L₂-projection body, L_p-intersection body,mixed intersection bodies,mixed new geometry bodyΓ_p,iK and a new notion from our definition—L_p—affine surface area,L_p-mixed quermassintegrals and L_p-mixed affine surface areas in the L_p-Brunn-Minkowski theory.In Chapter two we use the special property thatΠ₂K is an origin-centered ellipsoid to give two versions for the reverses of the L₂-Petty projection inequality(L₁-Petty projection inequality is just the Petty's conjectured projection inequality),at the samr time, the inclusive relationships between L₂-projection bodyΠ₂K and the classical projection bodyΠK are established,a contrained minimization problem is solved.In Chapter three,associated with the notions of Gardner's and Giannoponlos',or V.Yaskin's and M.yaskin's,or C.haberl's and M.Ludwig's,we reasonablly rewrite it usng some different signs—L_p-intersection body I_pK.On this geometry body,linearly equivalent property and several monotonicity results when p≤-1 are given,and together L_p-intersection body with Lp dual mixed quermassiintegrals(?)_-p,i(K,L),under the normalized L_p radial addition and L_p radial linear combination,we respectively show the dual quermassintegrals version's Brunn-Minkowski inequality and its isolate forms about L_p-intersection body.Lutwak extended Winterniz monotonicity problem,that is,if K∈Kⁿ and E is an ellipsoid,then if the areas of the projections of K do not exceed those of E,it follows thatΩ(K)≤Ω(E).Chapter four use an equivalent relationship between L_p-mixed volumes and L_p-extended affine surface areas,we extend Lutwak's result to L_p analog. As applications of this approach,we establish the L_p-mixed quermassintegrat version's Aleksandrov's projection theorem and Petty-Schneider theorem.In the context of convex geometry,the polar of a convex body is an important object, however,the polar of a star body may not exist.In Chapter five,by the notion of star dual of a star body that Moszynska introduced,and associated with star dual of mixed intersection bodies and dual quermassintegrals,with the harmonic p-combination and p-radial linear combination,we respectively state the dual quermassintegrals version's Brunn-Minkowski inequality about star dual of mixed intersection bodies.Recently,Lutwak,Yang and Zhang posed the notion of new geometric bodyΓ_-pK, in Chapter six,we introduce a new notion—the mixed new geometry bodyΓ_-p,iK(then new geometric body being its a special case).For this geometric body,we obtain five properties of the opertorΓ_-p,i,establish the relationship between the volumes ofΓ_-p,iK and that of convex body K,and study Shephard version's problem about the mixed new geometric body.As an aside,in Chapter seven,the concepts of the ith L_p-mixed anne surface area and L_p-polar curvature images are introduced,together L_p-centroid body with p-Blaschke linear combination,we prove the generalization of L_p-versions of the Busemann-Petty affine centroid inequality,and get the inequality analogous to the Blaschke-Santalóinequality for the L_p-mixed anne surface area and the general forms of the dual Urysohn inequality,and obtain the similar to the inequality for Marcus-Lopes,Bergstrom and Ky Fan.