Rotationally symmetric solutions to the Lp-Minkowski problem

In this paper we study the Lp-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 Lp-Minkowski problem which corresponds to the super-critical case of the Blaschke–Santalo inequality.