Product property for capacities in $\mathbb{C}^N$
Volume 106 / 2012
Abstract
The paper deals with logarithmic capacities, an important tool in pluripotential theory. We show that a class of capacities, which contains the $L$-capacity, has the following product property: $$C_\nu (E_1\times E_2)=\min (C_{\nu_1}(E_1),C_{\nu_2}(E_2)),$$ where $E_j$ and $\nu_j$ are respectively a compact set and a norm in $\mathbb C^{N_j}$ ($j=1,2$), and $\nu$ is a norm in $\mathbb C^{N_1+N_2}$, $\nu =\nu_1\oplus_p\nu_2$ with some $1\leq p\leq\infty$.
For a convex subset $E$ of $\mathbb C^N\!\!$, denote by $C(E)$ the standard $L$-capacity and by $ \omega_E$ the minimal width of $E$, that is, the minimal Euclidean distance between two supporting hyperplanes in $\mathbb R^{2N}$. We prove that $C(E)={\omega_E}/{2}$ for a ball $E$ in $\mathbb C^N$, while $C(E)={\omega_E}/{4}$ if $E$ is a convex symmetric body in $\mathbb R^N$. This gives a generalization of known formulas in $\mathbb C$. Moreover, we show by an example that the last equality is not true for an arbitrary convex body.
