  The dual Minkowski problem in the two-dimensional plane is studied in this
paper. By combining the theoretical analysis and numerical estimation of an
integral with parameters, we find the number of solutions to this problem for
the constant dual curvature case when $0<q\leq4$. An improved nonuniqueness
result when $q>4$ is also obtained. As an application, a result on the
uniqueness and nonuniqueness of solutions to the $L_p$-Alexandrov problem is
obtained for $p<0$.
