Let $\Omega$ be a proper open cone in a real Banach space $V$. We show that the tube domain $V \oplus i\Omega$ over $\Omega$ is biholomorphic to a bounded symmetric domain if and only if $\Omega$ is a normal linearly homogeneous Finsler symmetric cone, which is equivalent to the condition that $V$ is a unital JB-algebra in an equivalent norm and $\Omega$ is the interior of $\{ v^2: v\in V\}$.
Categories: math.DG