Inequality Of The Mixed Logarithmic John Ellipsoid

Convex geometry analysis is a modern geometry subject which mainly studies the geometric structure and invariants of geometric bodies(mainly convex bodies and stars)by using geometry and analytic methods.The classical Brunn-Minkowski theory is one of the most important theory in convex geometric analysis.Its devel-opment mainly includes the following several stages:the classical Brunn-Minkowski theory?Lp Brunn-Minkowski theory?Orlicz Brunn-Minkowski theory.The most important geometric body in convex geometry except sphere is ellipsoid,which is contained in the Brunn-Minkowski theory.In this paper,we generalize the mixed volume of L0.We define the standard Lp mixed volume of two convex bodies by the mixed cone-volume measure.By the mixed volume,we consider the existence and uniqueness of the solution of the extreme value problem(problem S0 and S0)for two convex bodies,thereby introducing the mixed logarithmic John ellipsoid.In this paper,we prove that the mixed cone volume measure satisfies the subspace concentration inequality(Theorem 3.1.6)under the condition that both convex bodies satisfy potential-like and both are symmetric about the origin.Secondly,we prove the uniqueness of the solution of the two extremum problems by discussing the isotropic property of the mixed cone volume measure(Theorem3.2.2).Finaly,we redefines the first two extremum problems and introduces the concept of the mixed logarithmic John ellipsoid.We also obtain the conclusion of the mixed logarithmic John ellipsoid and mixed cone volume measure satisfies the subspace concentration inequality under the con-dition that both convex bodies satisfy inflation and its centroid at the origin.