Barriers in metric spaces

✍ Scribed by Andreas W.M. Dress; Vincent Moulton; Andreas Spillner; Taoyang Wu

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 405 KB
• Volume: 22
• Category: Article
• ISSN: 0893-9659
• DOI: 10.1016/j.aml.2008.10.006

✦ Synopsis

Defining a subset B of a connected topological space T to be a barrier (in T ) if B is connected and its complement T -B is disconnected, we will investigate barriers B in the tight span

of a metric D defined on a finite set X (endowed, as a subspace of R X , with the metric and the topology induced by the ∞ -norm) that are of the form

for some f ∈ T (D) and some ε ≥ 0. In particular, we will present some conditions on f and ε which ensure that such a subset of T (D) is a barrier in T (D). More specifically, we will show that B ε (f ) is a barrier in T (D) if there exists a bipartition (or split) of the ε-support supp ε (f ) := {x ∈ X : f (x) > ε} of f into two non-empty sets A and B such that f (a) + f (b) ≤ ab + ε holds for all elements a ∈ A and b ∈ B while, conversely, whenever B ε (f ) is a barrier in T (D), there exists a bipartition of supp ε (f ) into two non-empty sets A and B such that, at least, f (a) + f (b) ≤ ab + 2ε holds for all elements a ∈ A and b ∈ B.