Definition 4

For any set A of a topological vector space E, the intersection of all the closed convex sets containing A is called the convex closed envelope of A; it is the smallest convex closed set containing A.

From prop. 14, the convex closed envelope of A is the closure of the convex envelope of A; it is clearly the same as the convex closed envelope of $\overline{A}$.

Similarly we call the smallest symmetric, convex, closed set that contains A, the symmetric convex closed envelope (or the balanced convex closed envelope) of A; it is the closure of the symmetric convex envelope of A (II, p. 10); it is also the symmetric convex closed envelope of $\overline{A}$.