  We establish curvature estimates for anisotropic Gauss curvature flows. By
using this, we show that given a measure $\mu$ with a positive smooth density
$f$, any solution to the $L_p$ Minkowski problem in $\mathbb{R}^{n+1}$ with $p
\le -n+2$ is a hypersurface of class $C^{1,1}$. This is a sharp result because
for each $p\in [-n+2,1)$ there exists a convex hypersurface of class
$C^{1,\frac{1}{n+p-1}}$ which is a solution to the $L_p$ Minkowski problem for
a positive smooth density $f$. In particular, the $C^{1,1}$ regularity is
optimal in the case $p=-n+2$ which includes the logarithmic Minkowski problem
in $\mathbb{R}^3$.
