Several Problems Of Generalized Brunn-Minkowski Theory

The Brunn-Minkowski theory is a main ingredient of convex geometric analysis,as the Brunn-Minkowski inequality and the isoperimetric problem are the core of the BrunnMinkowski theory.This thesis is devoted to the research on the Brunn-Minkowski inequality and the isoperimetric problem.We investigate the generalized Brunn-Minkowski inequality and its meanings,as well as the Lp John ellipsoid for negative p.As the first aspect of this thesis,we first study the ?-Brunn-Minkowski inequality for polar bodies of a given convex bodies with the origin in its interior.This is a new class of generalized Brunn-Minkowski inequalities which can unify the log-,the Orlicz-,the Lp-(p ?(0,1))Brunn-Minkowski inequalities,etc.An equivalence between the polar?-Brunn-MInkowski inequality and the polar ? Minkowski mixed volume inequalities has also been proved.These results show that the log-Brunn-Minkowski inequality is the strongest one among all known generalized Brunn-Minkowski inequalities.The second goal of this thesis is the John type ellipsoids arisen form the isoperimetric theory.The John ellipsoid is the one with biggest volume among all ellipsoids that contained in a given convex body.Its dual is called the Lowner ellipsoid,which is the one with smallest volume among all ellipsoids that contain the involved convex body.In 2005,Lutwak Yang,Zhang introduced the notion of the Lp John ellipsoid associated with a convex body for p>0.Later,they characterized the limited case where p tends to 0.In this thesis,we consider the case where p<0.Since the Lp mixed volume is not well defined for p<0,we turn to the mixed Lp affine surface area of convex bodies.Based on this,we find the characterization of the Lp John ellipsoid for p<0 when the involved convex body is smooth enough.In particular,we show that the Lp John ellipsoid is exactly the Lowner ellipsoid when p?-?.These results present a fruitful contribution to the generalized Brunn-Minkowski theory.It may help some sort of progress in the field of the conjecture on the log-BrunnMinkowski inequality and reverse isoperimetric problem.