Lemma 4.3 is missing some condition

[DungMinhDao]

In Lemma 4.3, the easy direction (Exercise 29) can easily be proved. However, the other direction needs additional conditions, otherwise, it's not correct. For example, consider f(x) = -√x for x ≥ 0. The function and the domain are convex, but there is no subgradient at 0

[adityavarre]

True, thanks for the nice catch. We should change the lemma conditions such that the sub-gradient is non-empty in the interior points of the domain as done in (Theorem 3.1.13 in Nesterov, 2004). We will update this.