The $L_p$-Minkowski problem with super-critical exponents

Abstract

The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in ${\mathbb{R}}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. Existence of solutions has been obtained in the sub-critical case $p>-n-1$, but the problem remains open in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids. Our results show that the $L_p$-Minkowski problem admits a solution in both the sub-critical and super-critical cases but does not have a solution in general in the critical case.