Suppose we've picked two points randomly from a uniform distribution over the Euclidean plane and we know that the Euclidean distance between them is $d$. What is the expected value of the Manhattan distance, $m$, between the two points, $E[m|d]$?
For context: I originally thought it was just a simple application of the Pythagorean theorem. But knowing that $a^2+b^2=c^2$ doesn't allow me to recover $a+b$ as a function of $c$, right? That function depends on factors other than the side lengths, and it's not immediately obvious to me as to how to proceed. For further context, I have haversine distances between pairs of points. I'd love a quick and dirty way to convert these to driving distances, for which Manhattan would work fine assuming a grid-like transportation network.