The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem

For solutions to the Minkowski problem of the even dual curvature measures ${\tilde{C}}_q$ in ${\mathbb{R}}^n$, $n\ge 2$, we prove the non-uniqueness for $q>n$ and $n\ge 2$, and the uniqueness for $0<q<n$ and $n=2$. These results are governed by our established logarithmic Brunn-Minkowski inequality for dual quermassintegrals ${\tilde{V}}_q$.