The extension of the concept of convexity closure we devised at a prior entry has some of the usual properties of standard convexity. In the following we consider a generic metric space S, and use the following notation:

CX := X \ S,

X ≤ Y := X is included in Y,

X ≥ Y := X is a superset of Y.

Also, we drop parentheses in some expressions to make them more readable.

Theorem. H_{wave} verifies the following properties (proof trivial):

- H
_{wave}(Ø) = Ø. - X ≤ H
_{wave}(X). - X ≤ Y → H
_{wave}(X) ≤ H_{wave}(Y).

Lemma. For any sets X, Y ≤ S, r ≥ 0, the following properties hold:

- B
_{r}(X U Y) = B_{r}(X) U B_{r}(Y). - B
_{r}(X ∩ Y) ≤ B_{r}(X) ∩ B_{r}(Y). - CB
_{r}CB_{r}CB_{r}CB_{r}X = CB_{r}CB_{r}X.

Proof. 1 and 2 are trivial. As for 3, an element x belongs to CB_{r}CB_{r}X iff B_{r}x ≤ B_{r}X, hence B_{r}CB_{r}CB_{r}X ≤ B_{r}X → CB_{r}CB_{r}CB_{r}X ≥ CB_{r}X → B_{r}CB_{r}CB_{r}CB_{r}X≤ B_{r}CB_{r}X → CB_{r}CB_{r}CB_{r}CB_{r}X ≤ CB_{r}CB_{r}X (the converse inclusion is trivial).

For H_{wave} to qualify as a closure operator it must be idempotent, i.e. H_{wave}(H_{wave}(X)) = H_{wave}(X), so that we can define convex sets as those in the range of H_{wave}. This property is not verified in general; as a counterexample take S = N \ {3} with dist(n,m) = |m − n| and X = {0}, resulting in H_{wave}(X) = {0,1}, H_{wave}(H_{wave}(X)) = {0,1,2}. So, we must impose some restrictions on S to guarantee the idempotence of H_{wave}.

Definition. S is regular if B_{r}X ≤ B_{r}Y → B_{s}X ≤ B_{s}Y for all X, Y in S and r ≤ s.

Lemma. If S is regular, CB_{r}CB_{r}CB_{s}CB_{s}X ≤ CB_{max{r,s}}CB_{max{r,s}}X for all X in S.

Proof. Let t = max{r,s}. x belongs to CB_{s}CB_{s}X iff B_{s}x ≤ B_{s}_{}X, which by the regularity of S implies that B_{t}x ≤ B_{t}X, hence CB_{s}CB_{s}X ≤ CB_{t}CB_{t}X and CB_{r}CB_{r}CB_{s}CB_{s}X ≤ CB_{r}CB_{r}CB_{t}CB_{t}X. By an equivalent argument, CB_{r}CB_{r}CB_{t}CB_{t}X ≤ CB_{t}CB_{t}CB_{t}CB_{t}X, and this latter set is CB_{t}CB_{t}X.

Theorem. If S is regular, H_{wave}(H_{wave}(X)) = H_{wave}(X) for all X in S.

Proof. Using the lemmas stated above and de Morgan's Laws, we have H_{wave}(H_{wave}(X)) =

= U_{r≥0}CB_{r}CB_{r}U_{s≥0}CB_{s}CB_{s}X =

= U_{r≥0}CB_{r}CU_{s≥0}B_{r}CB_{s}CB_{s}X =

= U_{r≥0}CB_{r}∩_{s≥0}CB_{r}CB_{s}CB_{s}X ≤

≤ U_{r≥0}C∩_{s≥0}B_{r}CB_{r}CB_{s}CB_{s}X =

= U_{r≥0}U_{s≥0}CB_{r}CB_{r}CB_{s}CB_{s}X ≤

≤ U_{r,s≥0}CB_{max{r,s}}CB_{max{r,s}}X = H_{wave}(X).

The converse inclusion is given by property 2 of H_{wave}.

We define convexity in the customary manner: X is convex if X = H_{wave}(X). Remember that this definition of convexity does not exactly coincide with the classical one for S = R^{n}.

- Ø and S are convex.
- The intersection of a familiy of convex sets of S is also convex.
- (S is regular) H
_{wave}(X) is convex for any arbitrary X in S.

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