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?