On the discrete logarithmic Minkowski problem (Q2801965)

If \(K\subset{\mathbb R}^n\) is a convex body (compact and convex set with non-empty interior) containing the origin as an interior point, the cone-volume measure of \(K\) is the Borel measure on the unit sphere \(S^{n-1}\) defined by NEWLINE\[NEWLINE V_K(\omega)=\frac{1}{n}\int_{x\in\nu_K^{-1}(\omega)}x\cdot\nu_K(x)d\mathcal{H}^{n-1}(x), \quad \text{for each Borel }\omega\subset S^{n-1}. NEWLINE\]NEWLINE Here \(\nu_K\) is the Gauss map of \(K\) (defined on the set of boundary points of \(K\) having a unique outer unit normal) and \(\mathcal{H}^{n-1}\) is the (\(n-1\))-dimensional Hausdorff measure.NEWLINENEWLINEThen, the logarithmic Minkowski problem asks for the necessary and sufficient conditions on a finite Borel measure \(\mu\) on \(S^{n-1}\) to be the cone-volume measure of an \(n\)-dimensional convex body. The logarithmic Minkowski problem is the relevant limit case \(p=0\) in the general \(L_p\)-Minkowski problem, introduced by \textit{E. Lutwak} in [J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)]. In the recent years, many important contributions to this problem have been obtained.NEWLINENEWLINERegarding the discrete case of the logarithmic Minkowski problem, it has been recently proved that any discrete measure on the sphere, whose support is in general position and not concentrated on a closed hemisphere, is a cone-volume measure [\textit{G. Zhu}, Adv. Math. 262, 909--931 (2014; Zbl 1321.52015)].NEWLINENEWLINEIn the paper under review the authors establish a new sufficient condition for the existence of a solution for the logarithmic Minkowski problem. More precisely, they prove that if \(\mu\) is a discrete measure on \(S^{n-1}\) which is not concentrated on any closed hemisphere and satisfies the essential subspace concentration condition, then \(\mu\) is the cone-volume measure of a polytope in \({\mathbb R}^n\) containing the origin in the interior.NEWLINENEWLINEHere, a finite Borel measure \(\mu\) on \(S^{n-1}\) is said to satisfy the essential subspace concentration condition if for every essential subspace \(\xi\) of \({\mathbb R}^n\) with respect to \(\mu\) (i.e., such that \(\xi\cap\text{supp}(\mu)\) is not concentrated on a closed hemisphere of \(\xi\cap S^{n-1}\)), \(0<\dim\xi<n\), the following conditions hold: {\parindent=6mm \begin{itemize}\item[(1)] \(\mu(\xi\cap S^{n-1})\leq(\dim\xi/n)\mu(S^{n-1})\), and \item[(2)] if equality holds for some \(\xi\), then there exists a subspace \(\xi'\), complementary to \(\xi\) in \({\mathbb R}^n\), such that \(\mu(\xi'\cap S^{n-1})=(\dim\xi'/n)\mu(S^{n-1})\). NEWLINENEWLINE\end{itemize}} The paper concludes with a section where some new inequalities for the cone-volume measures are established.