Curvature bound for Lp Minkowski problem

Abstract

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 Lp Minkowski problem in Rn+1 with p≤−n+2 is a hypersurface of class C1,1. This is a sharp result because for each p∈[−n+2,1) there exists a convex hypersurface of class [Formula presented] which is a solution to the Lp Minkowski problem for a positive smooth density f. In particular, the C1,1 regularity is optimal in the case p=−n+2 which includes the logarithmic Minkowski problem in R3.