\[LOCMAX\] and $\bar{\omega}, \bar{\lambda}\in U.$ Let $\bar{\omega } \notin U, \bar{\lambda} \in U.$ We show $\Psi(\bar{\omega}, \bar{\lambda}, \bar{a}, \bar{c}) \le 3^{1-\gamma} - \delta$ for some $\bar{a}, \bar{c}.$ This implies the claim as in the proof of Corollary \[SUMOPT\].

Let $\varepsilon'<\varepsilon/3$ and $U'= {\cal U}_{\varepsilon'}(1/3, 1/3).$ For $ \bar{\lambda} \notin U'$ the claim follows with Theorem \[OPT\] (2). For $ \bar{\lambda} \in U'$ we show that $\Psi(\bar{\omega}, \bar{\lambda}, \bar{a}, \bar{c}) \le 3^{1- \gamma}- \delta$ for $a_i=s$ and $ c_0=1, R(c_1, c_2)=(k \lambda_1, k \lambda_2).$ (Recall $\bar{\lambda} \in U.$) For $\Psi:= \Psi(\bar{\lambda}, \bar{\lambda}, \bar{a}, \bar{c})$ with $\bar{a}, \bar{c}$ as required by Lemma \[LOCMAX\] we have $\Psi \le 3^{1-\gamma}.$ Note, $Q(a_i)=\lambda_ik\gamma/\lambda_i= k \gamma$ which implies $a_i=s$ and $c_0=1, R(c_1, c_2)=  (k \lambda_1, k \lambda_2).$ Therefore all $a_i-$terms cancel and $\Psi= \prod (1/\lambda_i)^{\lambda_i}\cdot \left(\prod (\lambda_i/c_i)^{\lambda_ik\gamma}\right)p(\bar{c})^\gamma \le 3^{1-\gamma}. $

As $\bar{\omega} \notin U$ whereas $\bar{\lambda} \in U'$ and $\varepsilon' \le \varepsilon/3$ we have that $\prod (1/\omega_i)^{ \omega_i } \, \le \prod (1/\lambda_i)^{\lambda_i}- \delta'$ for a $\delta' >0$ (proof omitted.) Then $\Psi(\bar{\omega}, \bar{\lambda}, \bar{a}, \bar{c}) \le \Psi - \delta'\left(\prod 
(\lambda_i/c_i)^{\lambda_i k \gamma }\right)p(\bar{c})^\gamma .$ If $\Psi \le 3/2$, we are done. Otherwise we have that $\left(\prod_i(\lambda_i/c_i)^{\lambda_ik\gamma}\right)p(\bar{c})^\gamma $ is bounded below by $1/2$ (as $\prod (1/\lambda_i)^{\lambda_i} \le 