  Heinrich, Mankiewicz, Sims, and Yost proved that every separable subspace of
a Banach space $Y$ is contained in a separable ideal in $Y$. We improve this
result by replacing the term "ideal" with the term "almost isometric ideal". As
a consequence of this we obtain, in terms of subspaces, characterizations of
diameter 2 properties, the Daugavet property along with the properties of being
an almost square space and an octahedral space.
