On the \(L^p\) dual Minkowski problem for \(-1 < p < 0\) (Q6607674)

The article addresses the \(L_p\) dual Minkowski problem, a fundamental question in convex geometry. This problem seeks the existence of a convex body with a prescribed \((p,q)\)-th dual curvature measure \(\tilde{C}_{p,q}(K,\cdot)\) introduced by \textit{E. Lutwak} et al. [Adv. Math. 329, 85--132 (2018; Zbl 1388.52003)]. The study unifies two key topics in the field: the \(L_p\) Minkowski problem and the dual Minkowski problem.\N\NFocusing on the range \(-1 < p < 0\), \(q < p + 1\), and \(p \neq q\), the paper proves that for a given nonzero even finite Borel measure \(\mu\) on \(S^{n-1}\) there exists a(n origin symmetric) convex body \(K\in \mathbb{R^n}\) with \(\mu=C_{p,q}(K, \cdot)\) if and only if the given \(\mu\) is not concentrated on any lower-dimensional subspace. This somewhat completes the earlier picture, although the problem in the \((p<0, q>0)\) case remains unsolved in full generality.\N\NThe proof employs a variational approach, reformulating the problem into an optimization framework where the objective functional is maximized over origin-symmetric convex bodies. The optimizer is shown to correspond to the required convex body, thus solving the problem under the stated conditions. The result advances the \(L_p\) dual Brunn-Minkowski theory, offering new tools for studying geometric measures and related partial differential equations in convex geometry.