FractionalGeometry-Chap2

Write ha b 1 and hb c 2 a b1 24 b c2 25 and from

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ment holds for every point p ∈ (Aδ1 )δ2 , so (Aδ1 )δ2 ⊆ Aδ1 +δ2 . Proof of Prop. 2.2.1. We must show h satisfies positive-definiteness, symmetry, and the triangle inequality. Positive-definiteness Because ǫ-thickenings are defined only for ǫ ≥ 0, for all compact sets A and B , we see h(A, B ) ≥ 0. To show h(A, B ) = 0 if and only if A = B , first suppose h(A, B ) = 0. Then A ⊆ Bǫ and B ⊆ Aǫ for all ǫ > 0. It follows that A = B , because if not, then there is some a ∈ A − B , say. (Similar argument if it’s b ∈ B − A.) Compactness guarantees there is some ǫ > 0 for which d(a, b) ≥ ǫ for all b ∈ B , contradicting h(A, B ) < ǫ. To show A = B implies h(A, B ) = 0, suppose h(A, B ) = ǫ > 0. Then A ⊆ Bǫ and A Bǫ′ for all ǫ′ < ǫ. (Or the roles of A and B may be reversed. The argument is clear.) Because A and B are compact, there is a point a ∈ A with d(a, b) ≥ ǫ for all b ∈ B , and so A = B . Symmetry See Prob. 2.2.3. Triangle inequality Given any three comp...
View Full Document

This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.

Ask a homework question - tutors are online