The Gauss image problem with weak Aleksandrov condition (Q6611751)

Given a finite measure \(\lambda\) on the Lebesgue measurable subsets of the unit sphere \(S^{n-1}\) and a convex body \(K\subset{\mathbb R}^n\) containing the origin in the interior, let \N\[\N\lambda(K,\omega):= \lambda\left(S^{n-1}\cap \bigcup_{u\in\omega} N(K,\rho_K(u)u)\right),\quad\omega\subseteq S^{n-1} \mbox{ Borel},\N\]\Nwhere \(N(K,v)\) is the normal cone of \(K\) at \(v\in\partial K\) and \(\rho_K\) is the radial function of \(K\). If \(\mu\) is a finite Borel measure on \(S^{n-1}\), what are the necessary and sufficient conditions on \(\lambda\) and \(\mu\) so that there is a convex body \(K\) with \(o\in \mathrm{int}\,K\) and such that \(\mu=\lambda(K,\cdot)\)? This is the `Gauss image problem', suggested by \textit{K. J. Böröczky} et al. [Commun. Pure Appl. Math. 73, No. 7, 1406--1452 (2020; Zbl 1456.52002)]. They showed that the so-called Aleksandrov condition for \(\lambda\) and \(\mu\) is sufficient if \(\lambda\) is absolutely continuous (the solution body is unique up to a dilatation). Since the Aleksandrov condition is generally not necessary, the present author suggests a weaker form (implied by the Aleksandrov condition). The measure \(\mu\) is said to be weakly Aleksandrov related to \(\lambda\) if \(\mu(S^{n-1}) =\lambda(S^{n-1})\) and for each closed set \(\omega\subset S^{n-1}\), contained in a closed hemisphere, there exists \(\alpha\in (0,\pi/2)\) such that \(\mu(\omega)\le \lambda(\omega_{(\pi/2)-\alpha})\), where \(\omega_\beta\) is the spherical outer parallel set of \(\omega\) at distance \(\beta\). It is proved that this weak Aleksandrov condition is necessary and sufficient for a solution of the Gauss image problem if \(\lambda\) is absolutely continuous and \(\mu\) is discrete and not concentrated on a closed hemisphere. The proof uses maximization of the functional \N\[\N\Phi: K\mapsto \int_{S^{n-1}} \log \rho_K\,d\mu + \int_{S^{n-1}} \log \rho_{K^*}\,d\lambda\N\]\N(where \(K^*\) is the polar body of \(K\)) on polytopes. An important construction in this process is what the author calls partial rescaling of a polytope (moving some of its vertices on their rays). After the maximization, a variational result of Böröczky et al. [loc. cit.] can be used.