Research Of The Isoperimetric Problem And Related Extremal Theory

The research of this thesis belong to the theory of convex body functional analysis,Brunn-Minkowski theory,which is a high-speed developing geometry branch and is an important brand of geometric.This thesis devotes to the study of functional inequali-ties and extremal problems by using theory of geometry analysis.The researches refer to Loomis-Whitney inequality,isotropic measure,inequalities relating to L_p-version of Petty's conjecture projection inequality.These problems have attracted increased inter-est for this direction.This thesis mainly focuses on two issues,the first question is about the upper and lower bounds of the?n-1?-dimensional sectional volume of the reverse and dual Loomis-Whitney inequality.The second question is to establish inequalities of the p-surface area of projection body and L_pprojection body in L_pspace.In addition,we give the upper and lower bounds associated p-surface area with the volume of convex body.In chapter three,we establishing the reverse inequality of Meyer's,and estimate the?n-1?-dimensional sectional volume.By introducing a frame F={u₁,...,u_n},which is also the orthogonal basis in Rⁿ,and the definition of DLW-constant,we will prove a crucial lemma,and give the strict inequality and the proof of o-symmetric planar convex body.In addition,some conclusions about special convex bodies will be given,such as simplex and unit cube.Finally,we study the functional inequalities of convex body,and the estimate of the function will be given in this chapter.In chapter four,we illustrate the second question,we will establish the inequal-ity associate the projection body and L_pprojection body with p-surface area in L_p space.Based on our study of p-surface area,L_p-zonoid,combining with the L_p-Petty conjectured projection inequality,we establish the inequalities of p-surface area of L_p projection body.We also give the corresponding inequality if the convex body is L_p-zonoid.Finally,we estimate the upper and lower bounds of p-surface area,and extend the results of classic Brunn-Minkowski theory to L_pspace.