Hypersurfaces of Prescribed Curvature Measure
  We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on $\mathbb S^n$. 
Pengfei Guan, Junfang Li, and YanYan Li
math.DG math.AP