On The Loomis-whitney Type Inequality And Related Problems

The research content of this paper focuses on the Loomis-Whitney type inequality in convex geometry.The classical Loomis-Whitney inequality gives a relation between volume and projection measure of a convex body.Ball-Loomis-Whitney inequality,which is discrete form,is an extension of the classical Loomis-Whitney inequality.The Lp Loomis-Whitney inequality for even isotropic measures,which is a Lp-extension from discrete to continuous,is an important generalization of Ball-Loomis-Whitney inequality.This paper is devoted to establish the Loomis-Whitney type inequality for normalized Lp projection bodies and other problems related to projection bodies.This paper can be divided into three parts.In the first part,we define the mixed Lp surface area measure and establish the mixed Lp projection inequality.We construct the mixed Lp surface area measure for two convex bodies,and obtain the corresponding Lp,t mixed volume and its Lp-type Minkowski inequality.Then,we define the mixed Lp projection bodies and establish the mixed Lp projection inequality.In the second part,we establish the Loomis-Whitney type inequality for the normalized Lp projection bodies under the isotropic measure.Firstly,we give the lower bound of volume of the weighted Lp zonoid body.Then,from the estimation,we establish the Loomis-Whitney type inequality for the volume normalized Lp projection bodies,and give the conditions for equality to hold.In the third part,we study some L_?problems about convex bodies and inequalities for mixed bodies.Under the definition of L_?projection bodies and L_?centroid bodies,we can find that they have an inclusion relation,and then we can directly derive the L_?version of two important affine isoperimetric inequalities.Finally,from the definition and properties of mixed bodies,we derive some inequalities about mixed bodies.