On the \(L^{p}\) dual Minkowski problem for \(-1

The \(L^{p}\) $L^p$ dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the \(L^{p}\) $L^p$ dual Minkowski problem, asks about existence of a convex body with prescribed \(L^{p}\) $L^p$ dual curvature measure. This question unifies the previously disjoint \(L^{p}\) $L^p$ Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the \(L^{p}\) $L^p$ dual Minkowski problem for the case of \(q<p+1\) $q<p+1$ , \(-1<p<0\) $-1<p<0$ , and \(p\ne q\) $p\ne q$ for even measures.