Hypersurfaces of prescribed curvature measure

Abstract

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 a priori $C^2$-estimate for the corresponding curvature equation on ${\mathbb{S}}^n$.