Affine Isoperimetric Inequalities And Minkowski Problems For The Electrostatic Capacity

The Brunn-Minkowski theory of convex bodies centers around two kinds of interrelated problems:the isoperimetric type problems and Minkowski type problems.In this thesis,we establish some affine isoperimetric inequalities and study the Lp Minkowski problem for the electrostatic p-capacity.In Chapter 3,the Lp John ellipsoids for negative indices are studied.For convex bodies of class C+2 in Rn with the origin in their interiors,the existence of their Lp John ellipsoids for-n<p<0,as well as the non-existence of their Lp John ellipsoids for p<-n,is proved.For general convex bodies,we demonstrate the non-existence of their Lp John ellipsoids for p<0.In Chapter 4,the variation of the cone volume functional U introduced by LutwakYang-Zhang is derived.It becomes the first mixed volume of Minkowski when the convex body is strictly convex.An affine isoperimetric inequality for the variation of U is proved,which implies the LYZ conjecture for the functional U directly.Chapters 5 and 6 are devoted to the study of the Lp Minkowski problem for the electrostatic p-capacity for p≥n.In Chapter 5,a sufficient condition is given for the existence of solutions to the discrete measure case for 0<p<1.In Chapter 6,the existence and uniqueness of the solutions for p>1 are completely solved.