Plurisubharmonic geodesics and interpolating sets (Q2424598)

The authors apply the notion of geodesics of plurisubharmonic functions to the interpolation of compact subsets of \(\mathbb C^n\). Let \(K_0, K_1\subset \mathbb C^n\) be non-pluripolar compact sets which are polynomially convex, and let \(u_j\) be the relative extremal function of \(K_j\), for \(j=0,1\), with respect to some bounded hyperconvex domain \(\Omega\supset K_0\cup K_1\). Then for \(0<t<1\) the functions \(u_t\) define a geodesic between \(u_0\) and \(u_1\), where \[\begin{split} u_t(z)=\sup\Big\{\varphi(z,e^t)&:\, u\in \mathrm{PSH}\big(\Omega\times \{\zeta\in \mathbb C: 0<\log |\zeta|<1\}\big), \\&\limsup u(z,\zeta)\leq u_j, \forall \, z\in \Omega, \log|\zeta|\to j, j=0,1\Big\}. \end{split}\] The sets \[ L_t=\big\{z\in \Omega: u_t(z)=-1\big\}, \quad\ 0<t<1, \] converge in Hausdorff metric to \(K_j\), where \(t\to j\in \{0,1\}\). The authors prove that the interpolating sets \(L_t\) can be represented as sections \(K_t=\{Z: (z,e^t)\in \hat K\}\) of the holomorphic hull \(\hat K\) of the set \[ (K_0\times\{\zeta: \log |\zeta|=0\})\cup (K_1\times\{\zeta: \log |\zeta|=1\})\subset \mathbb C^{n+1} \] with respect to holomorphic functions in \(\mathbb C^n\times(\mathbb C\setminus \{0\})\). If in addition \(K_0,K_1\Subset\mathbb D^n\) are Reinhardt sets then the function \(t\mapsto \operatorname{cap}(K_t,\mathbb D^n)\) is logarithmically convex, where the Monge-Ampère capacity is defined by \[ \operatorname{cap}(K,\Omega)=\sup\big\{(dd^cu)^n(K): u\in \operatorname{PSH}^-(\Omega), \ u|_K\leq -1 \big\}. \] In particular the following Brunn-Minkowski inequality holds: \[ \operatorname{cap}(K_0^{1-t}K_1^t,\mathbb D^n)\leq \operatorname{cap}(K_0,\mathbb D^n)^{1-t}\operatorname{cap}(K_1,\mathbb D^n)^t, \quad 0<t<1. \] Equality occurs in the above inequalities for some \(t\in (0,1)\) if and only if \(K_0=K_1\).