Some Inequalities In Convex Bodies Geometry Of L_p-Space

This article belong to the domain of the L_p-Brunn-Minkowski theory(or called Brunn-Minkowski-Firey theory),a branch of Modern Geometric Analysis.Our main works are to research the basic 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.The main objects of our research are the inequalities and extremum properties for the volumes of some geometry bodies containing new ellipsoidΓ_-2K,the L_p-projection body,new geometry bodyΓ_-pK,radial mean body R_pK,mixed radial mean body R_p(K,L,μ),p-cross-section body C_pK,mixed p-cross-section body C_p(K,L),simplex and so on.The main results are as follows:1.A sufficient and neccessary condition is obtained about the polytope whose new ellipsoid is a ball.Some new properties are proven for operatorΓ_-2 and a set found in which the monotonicity of operatorΓ_-2 is held.As a special case,we study the new ellipsoid of polytope in detail.2.By putting an ellipsoid in its standard position,we proof some new properties and inequalities about L_p-projection bodies when they are balls.3.Some properties of p-radial mean bodies are generalized to mixed p-radial mean bodies.A few new properties of p-radial mean bodies is proven such as monotonicity, transform properties under volume preserving linear transformations.Using the polar coordinate formula for volume and Fubini's theorem,we obtain some inequalities reflecting relationships between V(K) and V(R_p(K,L,μ)).4.We also proof some new properties of p-cross-section bodies and define a new bodies—mixed p-cross-section for the first time and then generalize the properties of pcross-section bodies to mixed p-cross-section bodies.By studying the relationship bewteen p-mixed Radial mean bodies and mixed p-cross-section bodies we find underline relationships among p-cross-section bodies,projection bodies and radial mean bodies.5.We discussion a mass-point system in Eⁿ and put into more vigor to the relationships between the n-pseudo-symmetric point set and the simplex and then get a few sufficient and neccessary conditions for n-pseudo-symmetric point set.