On the Orlicz Minkowski problem for logarithmic capacity
In this paper, the Orlicz Minkowski problem for the logarithmic capacity is studied, and the existence of a solution for a general measure is proven by approximation. The Orlicz Minkowski problem on which the authors focus in this paper reads as follows. Let \(h_K\) denote the support function of the convex body \(K\), i.e., a compact convex subset of \(\mathbb{R}^n\) with non-empty interior, and let \(\mu_{\log}(K,\cdot)\) be the logarithmic capacitary curvature measure of \(K\) up to a constant (multiple of the surface area of the unit sphere \(S^{n-1}\) of the Euclidean space \(\mathbb{R}^n\)). Orlicz Minkowski problem for the logarithmic capacity. Given a suitable continuous function \(\varphi: (0,+\infty)\longrightarrow (0,+\infty)\) and a finite Borel measure \(\mu\) on the unit sphere \(S^{n-1}\), is there a convex body \(K\), such that for some \(c>0\), \[ c\varphi(h_K)\, \mathrm{d}\mu_{\log}(K,\cdot)=\mathrm{d}\mu? \] The measure \(\mu_{\log}(K,E)\), the logarithmic capacitary curvature measure of \(K\), is given by \[ \mu_{\log}(K,E)=\sigma_{n-1}^{-1}\int_E{g_*}(|\Delta u_K|^n\,\mathrm{d}S), \] where \(g_K\) is the Gauss map of \(K\). The main result of the paper is the following theorem. Theorem 1.1. Let \(\varphi : (0,+\infty)\longrightarrow (0,\infty)\) be a continuous, decreasing function with \(\lim_{x\to 0^+}\varphi(x)=+\infty\). Assume that \(\phi(x)=\int_0^x\,\frac{1}{\varphi(y)}\mathrm{d}y\) exists for any \(x>0\) and that it is unbounded as \(x\to+\infty\). Then, for any positive Borel measure \(\mu\) on the unit sphere \(S^{n-1}\) that is not concentrated on any closed hemisphere, there is a convex body \(K\) containing the origin such that \[ \mu=c\varphi(h_K)\mu_{\log}(K,\cdot) \] and \(\|h_K\|_{\phi,\mu}=1\), where \(c=\int_{S^{n-1}}{\frac{h_K}{\varphi(h_K)}}\,\mathrm{d}\mu\). From the latter, the existence of a solution to the \(L_p\) Minkowski Problem in [\textit{Z. Chen}, Potential Anal. 54, No. 2, 273--298 (2021; Zbl 1457.31008)] follows directly, as \(\varphi(x)=x^{1-p}\), for \(p>1\) satisfies the assumption in Theorem 1.1. To prove the main result, the authors provide, in first instance, the existence of a solution to the problem under the assumption that \(\mu\) is discrete and satisfies two conditions, A1) that it is not concentrated on a close hemisphere, and A2) that it does not have a pair of antipodal point masses. Using that for a finite positive Borel measure \(\mu\) on \(S^{n-1}\) satisfying condition A1), there exists a sequence of finite and discrete measures \(\{\mu_j\}\) on \(S^{n-1}\), which satisfy A1) and A2), and converges weakly to \(\mu\) [\textit{Z. Du} and \textit{X. Ge}, J. Differ. Geom. 116, No. 3, 555--596 (2020; Zbl 1453.31012)].
scientific article; zbMATH DE number 7472969
scientific article