Existence of Solutions to the Generalized Dual Minkowski Problem

Given a real number q and a star body in the n-dimensional Euclidean space, the generalized dual curvature measure of a convex body was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The corresponding generalized dual Minkowski problem is studied in this paper. By using variational methods, we solve the generalized dual Minkowski problem for \(q<0\), and the even generalized dual Minkowski problem for \(0\le q\le 1\). We also obtain a sufficient condition for the existence of solutions to the even generalized dual Minkowski problem for \(1<q<n\).