The definition of \(L_p\) dual curvature measures is based on the radial Gauss map and its reverse. Let \(K\subset{\mathbb R}^n\) be a convex body with \(o\in\mathrm{int}\,K\), let \(\rho_K\) denote the radial function of \(K\), defined on the unit sphere \(S^{n-1}\). The radial Gauss map \(\alpha_K\) associates to almost every \(u\in S^{n-1}\) the outer unit normal vector of \(K\) at \(\rho_K(u)u\). The reverse radial Gauss image \(\alpha_K^*(\eta)\) of a Borel set \(\eta\subseteq S^{n-1}\) is the set of all \(u\in S^{n-1}\) for which \(K\) has at \(\rho_K(u)u\) an outer normal vector falling in \(\eta\). Let \(p,q\in{\mathbb R}\) and let \(Q\subset {\mathbb R}^n\) be a star-shaped set containing \(o\) in the interior. The \(L_p\) \(q\)th dual curvature measure of \(K\) with respect to \(Q\) is defined by \[ \widetilde C_{p,q}(K,Q,\eta) =\frac{1}{n} \int_{\alpha_K^*(\eta)} h_K^{-p}(\alpha_K(u))\rho_K^q(u)\rho_Q^{n-q}(u)\,{\mathcal H}^{n-1}(du)\] for Borel sets \(\eta\subseteq S^{n-1}\), where \(h_K\) is the support function of \(K\) and \({\mathcal H}^{n-1}\) denotes the \((n-1)\)-dimensional Hausdorff measure. In this generality, these measures were introduced by E. Lutwak et al. [Adv. Math. 329, 85–132 (2018; Zbl 1388.52003)]. Special cases are the cone volume measure, Aleksandrov’s integral curvature measure (of the polar body), and the \(L_p\) surface area measure. A variational formula connects the measures \(\widetilde C_{p,q}(K,Q,\cdot)\) to generalized dual mixed volumes.
The present paper deals with Minkowski-type existence problems, asking for necessary and sufficient conditions in order that a given finite Borel measure on \(S^{n-1}\) arises as a measure \(\widetilde C_{p,q}(K,Q,\cdot)\) for a suitable convex body \(K\). In special cases, this amounts to solving a Monge-Ampère equation on \(S^{n-1}\). The authors find solutions for \(p>1\) and \(q>0\) (where they allow the origin to be on the boundary of \(K\)). Theorem 1 gives a solution for discrete measures. The proof uses an extremal problem and a variational argument. For general (finite) measures, it is known that the Minkowski problem must be modified, and Theorem 2 gives a solution for such a modified problem. The proof uses the discrete case and approximation. Theorem 3 provides regularity statements, based on results of Caffarelli. Theorem 4 shows, under special assumptions, that the solution is strictly convex. Theorem 5 improves Theorem 3 when \(o\) is an interior point of \(K\). Generally, the proofs require several technical subtleties.