Theorem. For any X in R^{n}

H_{wave}(X) = X U int(H_{convex}(X)),

where H_{wave}(X) = U_{r≥0}B_{r}(B_{r}(X) \ R^{n}) \ R^{n}, B_{r}(A) is the union of all closed balls of radius r centered at points in A, int(A) is the interior set of A and H_{convex}(X) is the convex hull of X.

Proof.

1. X U int(H_{convex}(X)) is included in H_{wave}(X): obviously X is included in H_{wave}(X). If x is in int(H_{convex}(X)), we can find an n-simplex σ around x that is entirely included in H_{convex}(X). From the definition of a convex hull, for each face f_{i} of σ there must be some x_{i} in X such that x_{i} lies in the interior of the half-space delimited by the f_{i} and not including x. Consider now an arbitrary ray h departing from x: h will intersect some face f_{i} in the manner depicted below.

Following the variable naming convention of the figure, if we set r(h) = (a^{2}+b_{i}^{2})/2a then for any point y in h dist(x_{i},y) > r(h) → dist(x,y) > r(h). It can be seen that the values r(h) are upper bounded by some r, as we have only finitely many b_{i}'s and a is bounded by the diameter of σ; and it follows that x belongs to B_{r}(B_{r}(X) \ R^{n}) \ R^{n}, since all points of R^{n} at a distance greater than r from every point of X must also be farther away than r from x.

2. H_{wave}(X) is included in X U int(H_{convex}(X)): Let x be a point outside X U int(H_{convex}(X)). By Hahn-Banach Separation Theorem, there is a hyperplane π separating x from int(H_{convex}(X)), which implies that all points of X are included at one of the closed half-spaces delimited by π. Let h be the ray normal to π with origin in π, passing through x and moving away from X.

For any r ≥ 0, the point y in h at distance r from x (in the direction away from X) verifies that dist(x',y) > r for every x' in X U int(H_{convex}(X)); this is so even in the case where x belongs to the boundary of H_{convex}(X) and hence lies directly on π. So for every r ≥ 0 x is in B_{r}(B_{r}(X) \ R^{n}) and thus does not belong to H_{wave}(X).

Corollary. cl(H_{convex}(X)) = ∩_{r}_{>0}H_{wave}(B_{r}(X)), where where cl(A) denotes the topological closure of A.

Proof. If x is in H_{convex}(X), then for any r > 0 x is also in int(B_{r}(H_{convex}(X)) = int(_{}H_{convex}(B_{r}(X)), which is included in H_{wave}(B_{r}(X)). Conversely, if x does not belong to cl(H_{convex}(X)) then by Hahn-Banach Separation Theorem there is a hyperplane separating x and cl(H_{convex}(X)), and a fortiori separating x and cl(H_{convex}(B_{r}(X))) for some r > 0; but the latter set is a superset of H_{wave}(B_{r}(X)), hence x does not belong to this either.

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