A priori bounds and existence of smooth solutions to Minkowski problems for log-concave measures in warped product space forms

In the present paper, we prove the a priori bounds and existence of smooth solutions to a Minkowski type problem for the log-concave measure $e^{-f(|x{|}^2)}dx$ in warped product space forms with zero sectional curvature. Our proof is based on the method of continuity. The crucial factor of the analysis is the a priori bounds of an auxiliary Monge-Ampère equation on ${\mathbb{S}}^n$. The main result of the present paper extends the Minkowski type problem of log-concave measures to the space forms and it may be an attempt to get some new analysis for the log-concave measures.