(in fact, this theorem is the exact analog to the theorem about ordered uniform random variables in the 1-dimensional case)

You could find the distribution of

given

. To that aim, for any

, you have:

and all these probabilities can be computed easily using annuli. For instance,

is the probability that there is 1 point in

and 9 points in

, and since these sets are disjoint, the numbers of points therein are independent (and Poisson distributed with parameter proportional to the area...).

Finally, you can use

to compute the expectation.

I let you try that.