  In [7], Dong and I proved that the domains $D \subset \mathbb{C}$ of finite
volume whose on-diagonal Bergman kernels $K(\cdot, \cdot)$ satisfy $K(z_0, z_0)
= Volume(D)^{-1}$ are disks minus closed polar sets. We utilized the solution
of the Suita conjecture, a deep theorem of several complex variables. In this
note, I present a significantly more elementary proof of this theorem that does
not use several complex variables. As a corollary, a new lower bound for the
on-diagonal Bergman kernel is given. Finally, I show that the only real
ellipsoid in Webster normal form which satisfies $K(0, 0) = Volume(D)^{-1}$ is
the unit ball.
