Example 2.2 (.3) - sample randomness

[Yarduza]

We stated that for Hoeffding's inequality to be valid, it's important that the sample from the "bin" will be random - the E(in).
In example 2.2.3 (Convex set, page 44), it's stated that we choose the sample data to be on the perimeter of a circle (as stated, we need to choose the N points carefully).

By choosing the N points that way, (or by using any other careful way), don't we mess with the randomness of the sample?
Is it possible that we can't use Hoeffding's inequality following this process at all?

[htlin]

The discussion of #dichotomies focuses on what the "worst" number of dichotomies is. Then, when data is sampled (as Hoeffding needs), the number of dichotomies would be no more than the worst case (as discussed with the growth functions). If we can manage to bound the growth functions, we can also bound the "actual # of dichotomies when data is sampled."

Hope this helps.