The concept of tight extensions of a metric space is introduced, the existence of an essentially unique maximal tight extension Tx-the "tight span," being an abstract analogon of the convex hull-is established for any given metric space X and its properties are studied. Applications with respect to (1) the existence of embeddings of a metric space into trees, (2) optimal graphs realizing a metric space, and (3) the cohomological dimension of groups with specific length functions are discussed. © 1984.
Dress, A. W. M. (1984). Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces. Advances in Mathematics. https://doi.org/10.1016/0001-8708(84)90029-X