Smooth solutions to the $$L_p$$ L p dual Minkowski problem

Abstract

In this paper, we consider the $$L_p$$ dual Minkowski problem by geometric variational method. Using anisotropic Gauss–Kronecker curvature flows, we establish the existence of smooth solutions of the $$L_p$$ dual Minkowski problem when $$pq\ge 0$$ and the given data is even. If $$f\equiv 1$$ , we show under some restrictions on p and q that the only even, smooth, uniformly convex solution is the unit ball.