Errors in Low and Lapsley's article "Optimization Flow Control, I: Basic Algorithm and Convergence"

In the note two errors in Low and Lapsley's article [1] are shown. Because of these errors the proofs of both theorems presented in the article are incomplete and some assessments are wrong. In the proof of Lemma 3 (at the beginning) on the page 871 it is written: " Given any p, q ≥ 0, using Taylor theorem and Lemma 2 we have ∇D(q)−∇D(p) = ∇ 2 D(w)(q −p) = [...] for some w = tp + (1 − t)q ≥ 0, t ∈ [0, 1]. " The problem is, that the mentioned Taylor theorem, or rather multidimensional mean value theorem, is not true. Here is the counterexample: