Barriers in metric spaces

Defining a subset BB of a connected topological space TT to be a barrier   (in TT) if BB is connected and its complement T−BT−B is disconnected, we will investigate barriers BB in the tight span T(D)={f∈RX:∀x∈Xf(x)=supy∈X(D(x,y)−f(y))} of a metric DD defined on a finite set XX (endowed, as a subspace of RXRX, with the metric and the topology induced by the ℓ∞ℓ∞-norm) that are of the form B=Bε(f)≔{g∈T(D):‖f−g‖∞≤ε}B=Bε(f)≔{g∈T(D):‖f−g‖∞≤ε} for some f∈T(D)f∈T(D) and some ε≥0ε≥0. In particular, we will present some conditions on ff and εε which ensure that such a subset of T(D)T(D) is a barrier in T(D)T(D). More specifically, we will show that Bε(f)Bε(f) is a barrier in T(D)T(D) if there exists a bipartition (or split  ) of the εε-support suppε(f)≔{x∈X:f(x)>ε} of ff into two non-empty sets AA and BB such that f(a)+f(b)≤ab+εf(a)+f(b)≤ab+ε holds for all elements a∈Aa∈A and b∈Bb∈B while, conversely, whenever Bε(f)Bε(f) is a barrier in T(D)T(D), there exists a bipartition of suppε(f) into two non-empty sets AA and BB such that, at least, f(a)+f(b)≤ab+2εf(a)+f(b)≤ab+2ε holds for all elements a∈Aa∈A and b∈Bb∈B.