L_p-Minkowski Problem

This thesis investigates the L_p-Minkowski problem for p=n1in n+1dimension-al Euclidean space, which corresponds to the critical exponent in the Blaschke-Santalo′inequality.We first consider the general case of this problem. Some basic but important prop-erties are reproved by diferent methods. And volume estimates for general solutions areobtained by analysis techniques.Based on this result, we then study the rotationally symmetric case of the L_p-Minkowski problem. A priori estimates for rotationally symmetric solutions is estab-lished by using a Kazdan-Warner type obstruction and blow-up analysis. Moreover,sufcient conditions for the existence of rotationally symmetric solutions are given bymethods such as calculus of variations, topological degree and approximation. This exis-tence result extends those in the case of n=1, which is all what people have known forthe existence of the L_p-Minkowski problem.Finally we include one existence result for another case of the L_p-Minkowski prob-lem, in which solutions are mirror-symmetric with respect to coordinate hyperplanes. Weneed to overcome difculties in dealing with several blow-up possibilities to obtain thisresult.