dist((x1,...,xn),(y1,...,yn)) := ∑|yi−xi|.
The picture displays several sets in Z2 (black cells) along with their wavefront hulls (black and gray cells). The resulting sets do not look at all like traditional convex sets (they do not even have to be connected), however they retain the operational properties of convexity (closed under intersections, etc.)
We investigate now how to calculate wavefront hulls in Zn.
Theorem. Zn is regular.
Proof. Let X, Y in Zn such that BrX ≤ BrY, s ≥ r. Since Btx = Bfloor(t)x for any radius t, we can assume without loss of generality that r and s are nonnegative integers. If x belongs to BsX \ BrX there exists some x' in X such that r < dist(x',x) = ∑|xi−x'i| = ∑σiδi ≤ s, where δi = xi−x'i, σi = sign(δi). It is easily seen that we can choose nonnegative integers η1,...,ηn such ηi ≤ σiδi and ∑ηi = r. If we define x'' with coordinates x''i = x'i + σiηi, we have dist(x',x'') = r, dist(x'',x) = dist(x',x) − r ≤ s − r, hence x belongs to Bs−rBrX. Since by hypothesis BrX ≤ BrY, we conclude that x also belongs to Bs−rBrY ≤ BsY.
Lemma. If a metric space S is regular, r ≤ s → CBrCBrX ≤ CBsCBsX for all X in S.
Proof. If x belongs to CBrCBrX then Brx ≤ BrX, and by the regularity of S Bsx ≤ BsX, i.e. x belongs to CBsCBsX.
Theorem. If X in Zn is bounded, Hwave(X) = CBrCBrX for some r ≥ 0.
This theorem allows us to devise a very primitive algorithm to calculate Hwave(X): traverse all the points of the minimun enclosing prism and compute for each point x whether Brx ≤ BrX with a large enough value of r (although this would need proof, r = diameter(X) seems to suffice). We will develop more sophisticated techniques at a later entry.