Research Of The Isoperimetric Problem And The Reverse Isoperimetric Problem Of Ellipsoid

The research of this thesis belong to the theory of Brunn-Minkowski theory,dual Brunn-Minkowski theory,and devoted to the study of extremal problems.This thesis devotes to the study of extremal problems and functional inequalities by using the theory of convex geometry analysis.This thesis mainly focuses on three questions,the first issue is the Mahler of the mixed LYZ ellipsoid.The second issue is the iteration inequalities of the Sine ellipsoid.The third issue is the volume inequalities of a new ellipsoid.In chapter 1,we give some background knowledge,research problems and achieve-ments of convex geometry.In chapter 2,we give some basic definitions,lemmas and common inequalities.In chapter 3,we improve the Mahler of the LYZ ellipsoid,and establish a sharp volume inequality for the mixed LYZ ellipsoid.Moreover,we obtain a sharp lower bounds of the volume product V(Γ_-2（K,L）)V（L^*）.By the definitions of the isotropic measureμ（·）:=n V_K,L（·）/h_L²（·）V_K,L(S^n-1)and the isotropic embedding,we prove a crucial lemma,and improve the unit map q:S^n-1→Sⁿis an isotropic embedding.Finally,using the prove method of the Mahler of the LYZ ellipsoid,we obtain sharp lower bounds of V(Γ_-2（K,L）)V（L^*）.In chapter 4,we establish volume inequalities of the Sine ellipsoid and the iteration inequalities about the sine ellipsoid operatorsΛ₂andΛ_-2.By the definitions of the sine ellipsoid,properties of the sine ellipsoid,the L_pBrunn-Minkowski theory and its dual theory,we establish the volume inequalities ofΛ₂K andΛ_-2K.Finally,we obtain the iteration inequalities ofΛ₂andΛ_-2.In chapter 5,we establish volume inequalities of a new ellipsoid.By the definitions of the LYZ ellipsoid and L_pdual curvature measure,we introduce a new ellipsoidΥ_-2,q（K,L）.Moreover,By Brunn-Minkowski inequality and dual Brunn-Minkowski inequality,we establish volume inequalities of new ellipsoid with the equality conditions.