We present new results on Kottman’s constant of a Banach space, showing (i) that every Banach space is isometric to a hyperplane of a Banach space having Kottman’s constant 2 and (ii) that Kottman’s constant of a Banach space and of its bidual can be different. We say that a Banach space is a Diestel space if the infimum of Kottman’s constants of its subspaces is greater that 1. We show that every Banach space contains a Diestel subspace and that minimal Banach spaces are Diestel spaces.

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