A unified approach used to generalize classical Brunn-Minkowski type inequalities to Lp Brunn-Minkowski type inequalities, called the Lp trans- ference principle, is refined in this paper. As illustrations of the effectiveness and practicability of this method, several new Lp Brunn-Minkowski type in- equalities concerning the mixed volume, moment of inertia, quermassintegral, projection body and capacity are established.

By using the method of mixed volumes, we give sharp bounds for inclusion measures of convex bodies in n-dimensional Euclidean space. In the special cases where the random convex body is the unit ball or when n = 3, neater and simpler bounds are obtained. All the associated inequalities proved are new isoperimetrictype inequalities.

The Orlicz–Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz– Legendre ellipsoids and isotropy of measures is demonstrated.

The Orlicz–John ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John ellipsoid and the evolved Lp John ellipsoids. The analog of Ball’s volume-ratio inequality is established for the new Orlicz–John ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John ellipsoids is demonstrated.

Convex bodies with identical John and LYZ ellipsoids are characterized. This solves an important problem from convex geometry posed by G. Zhang. As applications, several sharp affine isoperimetric inequalities are established.