TITLE: subsets of products of trees
QUESTION [7 upvotes]: A subset of a geodesic metric space is called convex if for every two points in the subset one of the geodesics connecting these points lies in the subset. Is it true that every convex subset of a product of two trees with $l_1$-metric is a median space, that is for every three points A,B,C in the subset there exists a point D in the subset such that D is on (some) geodesics connecting A and B, B and C, A and C?

REPLY [5 votes]: I simply wanted to add to the above that this is, I think, specific to a product of two trees, it is not true for at least three. Take $\mathbb{R}^3$ with the $l^1$ metric and inside it the plane $x+y+z=1$. It is convex (because Euclidean segments are geodesics also for $l^1$), not strongly convex, and not median: it contains the points $(1,0,0), (0,1,0), (0,0,1)$ but not their median point $(0,0,0)$.