This is an announcement for the paper “Ball Intersection Properties in Metric Spaces ” by Benjamin Miesch<https://arxiv.org/find/math/1/au:+Miesch_B/0/1/0/all/0/1>, Maël Pavón<https://arxiv.org/find/math/1/au:+Pavon_M/0/1/0/all/0/1>. Abstract: We show that in complete metric spaces, $4$-hyperconvexity is equivalent to finite hyperconvexity. Moreover, every complete, almost $n$-hyperconvex metric space is $n$-hyperconvex. This generalizes among others results of Lindenstrauss and answers questions of Aronszajn-Panitchpakdi. Furthermore, we prove local-to-global results for externally and weakly externally hyperconvex subsets of hyperconvex metric spaces and find sufficient conditions in order for those classes of subsets to be convex with respect to a geodesic bicombing. The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1610.03307