The Chen-Chvátal conjecture for metric spaces induced by distance-hereditary graphs

A special case of a theorem of De Bruijn and Erdős asserts that any noncollinear set of $n$ points in the plane determines at least $n$ distinct lines. Chen and Chvátal conjectured a generalization of this result to arbitrary finite metric spaces, with a particular definition of lines in a metric space. We prove it for metric spaces induced by connected distance-hereditary graphs -- a graph $G$ is called distance-hereditary if the distance between two vertices $u$ and $v$ in any connected induced subgraph $H$ of $G$ is equal to the distance between $u$ and $v$ in $G$.

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