Study On L_p-Brunn-Minkowski Theory

This Ph. D. dissertation sketches firstly the growing history, researching status, main represent figures in the researching branch and the author's research work; the following, it studies the monotone property of the generalized projection bodies , intersection bodies and centroid bodies; then, it studies emphasisly the quasi L_p-intersection bodies, the dual L_p-John ellipsoids, the isotropic L_p-surface area measure and so on.The main results given by the author are as follows:(i) The monotone property of the projection bodies intersection bodies and centroid bodies is the fundamental property in convex geometry. In fact, the monotone property of the projection bodies and the intersection bodies are the well known Shephard problem and the Busemann-Petty problem respectively. We established the monotone property for the generalized projection bodies intersecdtion bodies and centroid bodies. The generalized centroid bodies was first defined here.(ii) We defined the quasi L_p-intersection bodies and established the L_p-Busemann intersection inequality. We also obtained the dual Brunn-Minkowski inequality for the quasi L_p-intersection bodies. After generalized the notion of quasi L_p-intersection bodies to that of mixed quasi L_p-intersection bodies, we given the Aleksandrov-Fenchel inequality and an unique theorm.(iii) Given a convex body K, for p ≥ 1, we proved that there exists a family of ellipsoids E_pK such that the classical Lowner ellipsoid JK and the Legendre ellipsoid Γ₂K are the special cases of this family(p = ∞ and p = 2). This result is a perfect dual form of the 《L_p-John ellipsoids 》given by Lutwak, Yang and Zhang.(iv) Using the properties of L_p-John ellipsoids and dual L_p-John ellipsoids, we obtained a lot of isopermetric inequalities for L_p-projection bodies , L_p-intersection bodies, for example, incompletely exact forms of L_p-Petty projection inequality and the inverse form of the L_p-centroid inequality. Moreover, we got an inclusion of the Lp-John ellipsoids and using the John basis, we also obtained the L_p-analogs of Loomis-Whitney inequality and the Pythagorean inequality.