Existence of solutions to the generalized dual Minkowski problem (Q6611176)

Let \(q\) be a real number and \(Q\) be a star body in the \(n\) dimensional Euclidean space \(\mathbb{R}^n\) (that is, \(Q\) is star-shaped and its radial function is positive and continuous). Given a convex body \(K\) (a compact and convex subset of \(\mathbb{R}^n\), with non-empty interior), the generalized \(q\)-th dual curvature measure of \(K\) with respect to \(Q\), \(\tilde C(Q,K,\cdot)\), is defined by \N\[\N\tilde C(Q,K,\eta)=\frac1n\int_{\alpha^*_K(\eta)}\rho_K^q(u)\rho_Q^{n-q}(u)du, \N\]\Nfor every Borelsubset \(\eta\) of the unit sphere \(\mathbb{S}^{n-1}\) of \(\mathbb{R}^n\). Here \(\rho_K\) and \(\rho_Q\) denote the radial functions of \(K\) and \(Q\), respectively, \(du\) denotes the integration with respect to the standard uniform measure on \(\mathbb{S}^{n-1}\), and \(\alpha^*_K\) is the reverse radial Gauss image of \(K\). More precisely, for \(\eta\subset\mathbb{S}^n\), \N\[\N\alpha^*_K(\eta)=\{u\in\mathbb{S}^{n-1}\colon\rho_K(u)u\in\nu_K^{-1}(\eta)\},\N\]\Nwhere \(\nu_K\) is the Gauss map of \(K\). This class of measures was introduced by \textit{E. Lutwak} et al. [Adv. Math. 329, 85--132 (2018; Zbl 1388.52003)]. The corresponding Minkowski problem consists in finding necessary and sufficient conditions on a given measure \(\mu\) on \(\mathbb{S}^{n-1}\), such that there exists a convex body \(K\) verifying \N\[\N\tilde C_q(K,Q,\cdot)=\mu(\cdot).\N\]\NSeveral results concerning this problem are established in the paper, for different ranges of \(q\) and under distinct conditions on \(\mu\). These results contain either necessary and sufficient or just sufficient conditions for the existence of a solution. Let us mention two of them.\N\N\textbf{Theorem 1.} Let \(q\in(1,n)\) and let \(Q\) be an origin symmetric star body in \(\mathbb{R}^n\). If \(\mu\) is a finite, even Borel measure on \(\mathbb{S}^{n-1}\) which verifies the following condition \N\[\N\frac{\mu(\mathbb{S}^{n-1}\cap\xi)}{\mu(\mathbb{S}^{n-1})}<\min\left\{\frac iq,1\right\},\N\]\Nfor every \(i\)-dimensional subspace \(\xi\) of \(\mathbb{R}^n\), and for every \(i=1,\dots,n\), then there exists an origin symmetric convex body \(K\) in \(\mathbb{R}^n\), such that \N\[\N\tilde C_q(K,Q,\cdot)=\mu(\cdot).\N\]\N\textbf{Theorem 2.} Let \(Q\) be an origin symmetric star body in \(\mathbb{R}^n\), and let \(\mu\) be a finite, even Borel measure on \(\mathbb{S}^{n-1}\). There exists a convex body \(K\) in \(\mathbb{R}^n\), containing the origin in its interior, such that \N\[\N\tilde C_0(K,Q,\cdot)=\mu(\cdot),\N\]\Nif and only if \(\mu\) is not concentrated on any closed hemisphere of \(\mathbb{R}^n\).