Extremal Problems In Theory Of Convex Geometrical Functional Analysis

The researches of this thesis belongs to the theory of convex body functional analysis and Orlicz Brunn-Minkowski theory, and devoted to the study of functional inequalities and extremal problems. The thesis originates from the John theorem and is the subsequent researches of the generalizations of John theorem. This problems has attracted increased interest for this direction. The researches refer to Brascamp-Lieb inequality, Loomis-Whitney inequality, affine convex shell, Gauss-John position, isotropic measure, Orlicz Busemann-Petty centroid inequality and Orlicz John ellipsoids.We generalize a excellent work of Barthe:the multidimensional version of Brascamp-Lieb inequality and its reverse. Using the Cauchy-Binet formula, we will prove a crucial lemma. Then under the condition of positive multidimensional double John decomposi-tion, another form of geometric Brascamp-Lieb inequality and its reverse are obtained.The condition of John theorem is actually a discrete isotropic measure. Then as a natural generalization of it, we define an extended isotropic measure on the sphere. It is continuous on the sphere and can be characterized by the L-surface area of a convex body. We prove that the crucial Ball-Barthe lemma also holds for the positive extended isotropic measure. Therefore, adopting the mass transportation, we establish the continuous version of Brascamp-Lieb inequality and its reverse for the positive extended isotropic measure. As the applications of these inequalities, we extend the optimal volume inequalities for unit balls of subspaces of Lp.We obtain the k-dimensional Loomis-Whitney inequality and the continuous ver-sion of Loomis-Whitney inequality. These inequalities are the essential generalizations of Loomis-Whitney inequality.Applying the method of positive definite quadratic forms developed by Voronoi and Gruber, we characterize the minimum affine convex shell. Moreover, the convex shell with the positions of maximal volume and minimum volume will be investigated.Using the different approach of the optimization theorem of John, we investigate the extremal problems of minimal eP(TK), which satisfies the Gauss-John position. The distance between a convex body which is in the extremal position of minimal (ep(TK))p and the Euclidian unit ball is obtained. In chapter 6, we consider the problem of isotropic measure in a different view. We shall choose no canonical scalar product in Rn yet, but the unique scalar product associated to a centrally symmetric ellipsoid. We give a necessary and sufficient condition forε-isotropic measure which induced by this scalar product, and characterize the John theorem by it.In chapter 7, using shadow systems, we provide a new proof of the Orlicz Busemann-Petty centroid inequality, which was first obtained by Lutwak, Yang and Zhang [139]. It is one of the Orlicz affine isoperimetric inequality.Lp John ellipsoids, which were introduced by Lutwak, Yang and Zhang [136], include classical John ellipsoids, Petty ellipsoids and dual Legendre ellipsoids. Finally, As the natural extension of Lp John ellipsoids, Orlicz John ellipsoids will be defined. Moreover, some properties of Orlicz mixed volume and Orlicz John ellipsoids are investigated.