The Minkowski problem deals with existence, uniqueness, regularity and stability of closed convex hypersurfaces whose Gauss curvature (as a function of the outer normals) is preassigned. Many variants of the Minkowski problem have been studied.
In this article the authors consider the \(L_p\)-Minkowski problem: Suppose \(\alpha \in \mathbb R\) is fixed. Under what conditions on a measure \(\mu\) on \(\mathbb S^{n-1}\) does there exist a convex body \(K\) such that \[ h(K,.)^{\alpha}dS(K,.)=d\mu, \] where \(h(K,.)\) is the support function and \(S(K,.)\) is the surface area measure? Obviously for the case \(\alpha=0\) the \(L_p\)-Minkowski problem reduces to the classical Minkowski problem. A partial solution to the \(L_p\)-Minkowski problem (when \(\alpha \leq 0\), \(\alpha \neq 1-n\) and \(\mu\) is a Borel measure whose support is not contained in a great subsphere of \(\mathbb S^{n-1})\) was given by Lutwak in 1993.
In this paper the authors prove that if we normalize by the volume \(V(K)\) of the solution \(K\), then there is a solution to the even \(L_p\)-Minkowski problem for all \(\alpha \leq 0\) with no additional restriction. The normalized even \(L_p\)-Minkowski problem is equivalent to the even \(L_p\)-Minkowski problem for all \(\alpha\) except \(1-n\). The authors first present a solution to a discrete version of the problem. To prove existence they use tools provided by the Brunn-Minkowski-Firey theory and to see that the solution is unique they use the \(L_p\)-Minkowski inequality. Then the solution of the \(L_p\)-Minkowski problem with even data follows from the discrete case by approximation arguments. However, for the \(L_p\)-Minkowski problem new a priori estimates are required to show that the minimizing sequence is bounded from below as well as from above.