Recall that a space is $\delta$-hyperbolic if there is some number $\delta$ with the property that every point on an edge of a geodesic triangle lies within $\delta$ of another edge. For example a tree is $0$-hyperbolic. One of the basic facts about standard hyperbolic space is that it is $\delta$-hyperbolic for some $\delta$, and I am looking for the smallest delta which makes this true.

Full disclosure: I stole this question from Dima Burago, who brought it up as an example of of a useless problem about which he is nevertheless a little curious. I haven't exactly burned the midnight oil, but I can't solve it.