The Dual Minkowski Problem for p-Capacity

A new family of geometric Borel measures on the unit sphere is introduced, which extends the \(L_0\) $L_0$ p-capacitary measures proposed by Zou and Xiong (J Differ Geom 116:555–596, 2020). In this paper, we consider the existence of the solution to the p-capacitary dual Minkowski problem when \(1<p<n\) $1<p<n$ and \(q<0\) $q<0$ . Moreover, log-Minkowski style inequality for dual mixed p-capacity when \(q<0\) $q<0$ is established