Before revision:  On the other hand, using this iteration complexity we prove if for each p' \in conv(S) with p > \varepsilon that is not a witness there is v \in S with p' - v \geq \sqrt{1+ \varepsilon to O(1/\varepsilon).
After revision:  On the other hand,  p > \varepsilon that is not a witness there is v \in S with p' - v \geq \sqrt{1+ \varepsilon to O(1/\varepsilon).