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

A classical theorem of Euclidean geometry 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.