Abstract:
Given a pair of compact, non-pluripolar, polynomially convex subsets $K_0$, $K_1$ of a bounded hyperconvex domain $\Omega\subset\Bbb C^n$, we consider a plurisubharmonic geodesic $u_t(z)$, $0<t<1$, between the functions $c_j\, \omega_j(z)$, $j=0,1$, where $c_j$ are positive constants and $\omega_j$ are the extremal functions of the sets $K_j$ relative to $\Omega$: $\omega_j(z)=\sup\{u(z): u\in PSH(\Omega),\ u<0, u|_{K_j}\le-1\}$.
The sets $K_t=\{z\in\Omega: u_t(z)=\min_\Omega u_t\}$ interpolate $K_0$ and $K_1$. For a good choice of the constants $c_j$, the relative capacities $Cap\,(K_t,\Omega)$ are proved to satisfy a stronger version of Brunn-Minkowski type inequality. This is achieved by using linearity of the Monge-Ampère energy functional $\int_\Omega u_t(dd^c u_t)^n$.
When the sets $K_j$ are Reinhardt subsets of the unit polydisk, $K_t$ do not depend on the choice of the constants $c_j$ and are the geometric means of $K_0$ and $K_1$: $K_t=K_0^{1-t}K_1^t$, and their capacities are $n!$ times the covolumes of certain unbounded convex subsets of $\mathbb R_+^n$.
