On the \(L_{p}\) Monge-Ampère equation (Q2013921)

The authors deal with the \(L_p\) Monge-Ampère equation \(h^{1-p}\,\text{det}\,(h_{ij}+h\delta_{ij})=f\), \(0<p<1\), which is the smooth case of the \(L_p\) Minkowski problem. They solve it on the unit sphere \(S^{n-1}\). The Minkowski problem is about necessary and sufficient conditions on a finite Borel measure \(\mu\) so that \(\mu\) is the \(L_p\) surface measure of a convex body in \(\mathbb{R}^n\). When \(\mu\) has a density \(f\) with respect to the spherical Lebesgue measure, the \(L_p\) Minkowski problem is equivalent to the Monge-Ampère equation mentioned above. Hence the main results of the paper have to do with the measure \(\mu\) in the \(L_p\) Minkowski problem.