On the planar dual Minkowski problem

In this paper, we resolve the planar dual Minkowski problem, proposed by Huang et al. (2016) [31] for all positive indices without any symmetry assumption. More precisely, given any q>0, and function f on S1, bounded by two positive constants, we show that there exists a convex body Ω in the plane, containing the origin in its interior, whose dual curvature measure C˜q(Ω,⋅) has density f. In particular, if f is smooth, then ∂Ω is also smooth.