Title: Concave domains with trivial biholomorphic invariants
Abstract: It is proved that if $F$ is a convex closed set in ${\mathbb C}^n$, $n\ge 2,$ containing at most one $(n-1)$-dimensional complex hyperplane, then the Kobayashi metric and the Lempert function of ${\mathbb C}^n\setminus F$ identically vanish.