The Orlicz Minkowski Problem And Related Extremal Problems

This dissertation deals with topics in Orlicz-Brunn-Minkowski theory. This area is just newly emerged in2010by Lutwak, Yang and Zhang. This dissertation is devoted to Orlicz Minkowski problem and related extremal problems.In chapter2, we give the solution of the Orlicz Minkowski problem for gen-eral measures. This result generalizes the solution of the even Orlicz Minkowski problem obtained by Haberl, Lutwak, Yang, and Zhang. Minkowski problem was proposed by Minkowski in1897, asks which measures on the sphere are surface area measures? This problem is essentially to find the solution of the Monge-Ampere equation, and is very important in geometric analysis, remain the focus of the recent research trend. Different from the variational solution of the even Orlicz Minkowski problem, we first give the the solution of the Orlicz Minkowski problem for discrete measures, then use the approximation arguments to give the solution of the Orlicz Minkowski for general measures. Futhermore, we give the recently known solution of the Lp Minkowski problem (p>1).In chapter3, we use the result of Paouris and Pivovarov, give an asymmetric Orlicz centroid inequality for probability measures. Moreover, we generalize the asymmetric Lp centroid inequality due to Haberl and Schuster from star bodies to compact sets. The Orlicz centroid inequality for convex bodies was established by Lutwak, Yang, and Zhang. Later on, Zhu gave the Orlicz centroid inequality for star bodies. Paouris and Pivovarov established the symmetric Orlicz centroid in-equality for probability measures by studying the random polytopes. We observe that the result of Paouris and Pivovarov is actually a special case of the M-addition. The recent study of Gardner, Hug, and Wei yields that1-unconditional bodies and the bodies created by the intersection of1-unconditional bodies with the first octant playing an important role in the theories of symmetric and asym-metric, which inspires us to establish an asymmetric Orlicz centroid inequality for probability measures.In chapter4, we establish Gaussian inequalities for Wulff shapes. In the90s, seminal work of Ball and Barthe on the Brascamp-Lieb inequality for isotropic measures and its reverse form turned out to be groundbreaking in the field of reverse affine isoperimetric inequalities. In particular, the Ball-Barthe approach has been successfully applied in two parallel lines of research. One is the study of the volume inequalities, the other is the study of the Gaussian inequlities. For the latter, Barthe and Schmuckenschlager established the mean width inequality and its reverse, respectively. Li and Leng later gave the continuous versions of these inequalities. Motivated by volume inequalities for Wulff shapes due to Schuster and Weberndorfer, we establish new Gaussian inequalities for Wulff shapes such that all above mentioned results are their special cases. We also give new Gaussian inequalities related to LYZ ellipsoids.In chapter5, we introduce the concept of complex isotropic measures, and establish volume inequalities of the Lp-cosine transform and the sine transform for complex isotropic measures by using the multidimensional Brascamp-Lieb and its reverse. In real vector spaces, volume inequalities of the Lp-cosine transform for isotropic measures were due to Ball, Barthe, Lutwak, Yang and Zhang. Volume inequalities of the sine transform for isotropic measures were due to Maresch and Schuster. The results in Chapter5can be seen as the generalization of their results in complex vector spaces.