Affine Isoperimetric Inequality Of L_p Projection Mean Ellipsoid

Brunn-Minkowski theory originated from the research work of H.Brunn and H.Minkowski.It has a history of more than one hundred years and has gone through three development stages:the classical stage,theL_p stage and the Orlicz stage.This paper mainly study affine inequality problem underL_p Brunn-Minkowski theory,which belongs to the category of convex body extremum.The convex body extremum problem is usually presented in the form of geometric inequalities.Specific to the development of isoperimetric problems,there are classical isoperimetric inequality,extended isoperimetric inequality,Petty projection inequality and Lutwak conjecture.The Lutwak conjecture,an important problem in geometry,has made no progress in the past thirty years.A first step to attack the longstanding Lutwak conjecture by using projection mean ellipsoid in R~n was taken by D.Zou and G.Xiong.On top of their work,we extend the projection mean ellipsoid to the framework ofL_p Brunn-Minkowski theory and prove a new affine isoperimetric inequality by usingL_p projection mean ellipsoid.