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
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Write |
From |
From |
A similar argument shows |
From |
Return to Hausdorff distance.