  We use the Suita conjecture (now a theorem) to prove that for any domain
$\Omega \subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies
$K(z_0, z_0) = \hbox{Volume}(\Omega)^{-1}$ for some $z_0 \in \Omega$ if and
only if $\Omega$ is either a disk minus a (possibly empty) closed polar set or
$\mathbb{C}$ minus a (possibly empty) closed polar set. When $\Omega$ is
bounded with $C^{\infty}$-boundary, we provide a simple proof of this using the
zero set of the Szeg\"o kernel. Finally, we show that this theorem fails to
hold in $\mathbb{C}^n$ for $n > 1$ by constructing a bounded complete Reinhardt
domain (with algebraic boundary) which is strongly convex and not biholomorphic
to the unit ball $\mathbb{B}^n \subset \mathbb{C}^n$.
