The Dual Minkowski Problem for Positive Indices

Abstract We derive the stability result of the dual curvature measure with near-constant density in the even case. As an application, the existence and uniqueness of solutions to the even dual Minkowski problem for positive indices in $\mathbb R^{n+1}$ are obtained with $n\geq 1$, provided the density of the given measure is close to 1 in the $C^{\alpha }$ norm with $\alpha \in (0,1)$.