Rotationally symmetric solutions to the L _p-Minkowski problem

In this paper we study the L _p-Minkowski problem for p=-n-1, which corresponds to the critical exponent in the Blaschke-Santalo inequality. We first obtain volume estimates for general solutions, then establish a priori estimates for rotationally symmetric solutions by using a Kazdan-Warner type obstruction. Finally we give sufficient conditions for the existence of rotationally symmetric solutions by a blow-up analysis. We also include an existence result for the L _p-Minkowski problem which corresponds to the super-critical case of the Blaschke-Santalo inequality.