  We prove curvature estimates for general curvature functions. As an
application we show the existence of closed, strictly convex hypersurfaces with
prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only
assumed to be monotone, symmetric, homogeneous of degree 1, concave and of
class $C^{m,\al}$, $m\ge4$.
