  We revisit the (block-angular) min-max resource sharing problem, which is a
well-known generalization of fractional packing and the maximum concurrent flow
problem. It consists of finding an $\ell_{\infty}$-minimal element in a
Minkowski sum $\mathcal{X}= \sum_{C \in \mathcal{C}} X_C$ of non-empty closed
convex sets $X_C \subseteq \mathbb{R}^{\mathcal{R}}_{\geq 0}$, where
$\mathcal{C}$ and $\mathcal{R}$ are finite sets. We assume that an oracle for
approximate linear minimization over $X_C$ is given. In this setting, the
currently fastest known FPTAS is due to M\"uller, Radke, and Vygen. For $\delta
\in (0,1]$, it computes a $\sigma(1+\delta)$-approximately optimal solution
using $\mathcal{O}((|\mathcal{C}|+|\mathcal{R}|)\log |\mathcal{R}| (\delta^{-2}
+ \log \log |\mathcal{R}|))$ oracle calls, where $\sigma$ is the approximation
ratio of the oracle. We describe an extension of their algorithm and improve on
previous results in various ways. Our FPTAS, which, as previous approaches, is
based on the multiplicative weight update method, computes close to optimal
primal and dual solutions using $\mathcal{O}\left(\frac{|\mathcal{C}|+
|\mathcal{R}|}{\delta^2} \log |\mathcal{R}|\right)$ oracle calls. We prove that
our running time is optimal under certain assumptions, implying that no
warm-start analysis of the algorithm is possible. A major novelty of our
analysis is the concept of local weak duality, which illustrates that the
algorithm optimizes (close to) independent parts of the instance separately.
Interestingly, this implies that the computed solution is not only
approximately $\ell_{\infty}$-minimal, but among such solutions, also its
second-highest entry is approximately minimal. We prove that this statement
cannot be extended to the third-highest entry.
