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We develop a theory of isometric subgraphs of hypercubes for which a certain inheritance of isometry plays a crucial role. It is well known that median graphs and closely related graphs embedded in hypercubes bear geometric features that involve realizations by solid cubical complexes or are expressed by Euler-type counting formulae for cubical faces. Such properties can also be established for antimatroids, and in fact, a straightforward generalization ("conditional antimatroid") captures thisdoi:10.1016/j.ejc.2005.03.001 fatcat:xxk6gw2fufgexiuyaxzg5chfz4