Curvature bound for L Minkowski problem

We establish curvature estimates for anisotropic Gauss curvature flows. By using this, we show that given a measure μ with a positive smooth density f , any solution to the L p Minkowski problem in R n + 1 with p ≤ − n + 2 is a hypersurface of class C 1 , 1 . This is a sharp result because for each p ∈ [ − n + 2 , 1 ) there exists a convex hypersurface of class C 1 , 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 R 3 .