The Measure Comparison Problems And Related Topics

In this thesis, we mainly study the comparison of the Lebesgue measure and the arbitrary measure of the star cross section in n-complex space and the sharpness of the important analytical inequalities in the convex geometry.According to the properties of the intersections of star bodies and the pro-jections of the convex bodies in the Euclidean space, the complex situation was investigated by Koldobsky. Koldobsky’s results developed a new direction for the research of convex geometry. As a classical theorem in convex geometry, Funk’s section theorem plays important roles in volume comparison problems. Its complex correspondence is of essential interest in the topics about measure comparison prob-lem. However, to do so, it is quite different from the situation in the Euclidean space Rn. In fact, to extend the Funk’s section theorem to the n-complex space, more ef-fort is required. The main quest is the complex generalized intersection bodies. The difficulty for such an extension exists in that we can’t apply the dense property of the so-called generalized intersection bodies in origin symmetric star bodies to the n-complex situation. Therefore we can only establish the complex version of Funk’s section theorem to complex generalized intersection bodies. The second main result of this thesis is about the complex Funk’s section theorem for arbitrary measure. This is an analogy of the Busemann-Petty problem for arbitrary measure due to Zvavitch.Sharp affine Lp Gagliardo-Nirenberg inequality and the analytic version of the Blaschke-Santalo inequality are two important inequalities. In this thesis, we sharp them by establishing their affine forms. The main tools are general Lp projection bodies and general Lp moment bodies.