Extremum Problems In Convex Bodies Geometry Of L_p-Space

This article belong to the domain, 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). Our main works are to research the theories of convex bodies, some inequalities and extremum properties of geometry bodies by applying the basic notions, basic theories and integral transforms of the L_p-Brunn-Minkowski theory. In the first aspect, the basic theories of the L_p-Brunn- Minkowski theory are studied. The other parts, we research the inequalities and extremum properties of some geometry bodies containing the L_p-projection body, L_p-centroid body, new geometry body , L_p-John ellipsoid, L_p-curvature image and a new notion from our definition-L_p-mixed projection body for these volumes, quermassintegrals and affine surface areas in the L_p-Brunn-Minkowski theory.In the aspects of the basic theory for L_p-Brunn-Minkowski theory, we first show the dual of L_p-mixed quermassintegral -the notion of L_p-dual mixed quermassintegral which is the extensions of L_p-dual mixed volume and dual mixed quermassintegrals. For this new notion, we not only study its qualities, integral represntation and Minkowski inequality, but also establish the Brunn-Minkowski inequality of dual quermassintegrals for the Inharmonic radial combination of star bodies by applying this new notion, this inequality and the Brunn-Minkowski inequality of quermassintegrals for the Firey L_p- conbination of convex bodies appear "dual" form. For above two Brunn-Minkowski inequality, these isolates are respectively obtained. The reverse of the well-known Blaschke-Santalo inequality is researched in the classical Brunn- Minkowski theory to be interest, we unexpectedly get a reverse of the Blaschke-Santalo inequality by applying the monotonicity of the new geometry body Γ_-pK.We put into more vigor for the inequalities and extremum properties of geometry bodies in the L_p-Brunn-Minkowski theory. About the L_p-projection body, we mainly research the L_p-forms of the Petty's conjectured projection inequality (this is just the Petty's conjectured projection inequality when p = 1), and give several results, we particularly prove that the Petty's conjectured projection inequality is true when p = 2. In the meantime, we give several versions for the reverses of the L_p-Petty projection inequality. In addition, we also get better results for the monotonicity inequalities of the L_p-projection...