Convergence of Deterministic IFS

The triangle inequality for the Hausdorff distance is

h(A,B) ≤ h(A,C) + h(C,B)

for all compact sets A, B, and C.

The name comes from the familiar observation about Euclidean distances and the lengths of sides of triangles:

dist(A,B) ≤ dist(A,C) + dist(C,B)

First, note the Lemma (Ae)f ⊆ A(e+f).

Write h(A,B) = p, h(A,C) = q, and h(C,B) = r, so

A ⊆ Bp, B ⊆ Ap,   A ⊆ Cq, C ⊆ Aq,   C ⊆ Br, and B ⊆ Cr,

From C ⊆ Br we see Cq ⊆ (Br)q ⊆ Br+q, by the lemma.

From A ⊆ Cq and Cq ⊆ Br+q we see A ⊆ Br+q

A similar argument shows B ⊆ Ar+q. (Which two of the six containments above are used to show this?)

From A ⊆ Br+q and B ⊆ Ar+q we see h(A,B) ≤ q + r = h(A,C) + h(C,B).

Return to Hausdorff distance.