Gated sets in metric spaces

Abstract

Abstract. The concept of a gated subset in a metric space is studied, and it is shown that properties of disjoint pairs of gated subsets can be used to investigate projections in Tits buildings.

Metric fixed point theory on hyperconvex spaces: recent progress

“…(Remember that a convex set in a uniquely geodesic metric space is any set which contains any segment with endpoints in the same set.) Moreover, it was noticed in [18] that gated subsets of a complete geodesic space X are proximinal nonexpansive retracts of X. The next lemma is not hard to prove.…”

Decomposition andl1-Embedding of Weakly Median Graphs

“…An induced subgraph H of a graph G is called gated if for every vertex x outside H there exists a vertex x (the gate of x) in H such that each vertex y of H is connected with x by a shortest path passing through the gate x ; cf. [18]. G is a gated amalgam of two graphs G 1 and G 2 if G 1 and G 2 are (isomorphic to) two intersecting gated subgraphs of G whose union is all of G. A graph with at least two vertices is said to be prime if it is neither a proper Cartesian product nor a gated amalgam of smaller graphs.…”

The maximum multiflow problems with bounded fractionality

“…This roughly means that a shortest path from R to u can enter K p through the gate gR; our definition of the gate is compatible to that in [5]. The next lemma describes the gate of R when a shortest path from R to p enters K p via u ̸ = p.…”