Every minimax-optimal fixed-step method for Lipschitz convex minimization corresponds to a unique set of dual multipliers \(\lambda_{ij} \geq 0\), \(i < j\), sitting inside the polytope
\[ \mathcal{P}^N = \left\{\, \lambda \mid \lambda \geq 0,\ \ \sum_{i=j+1}^{N} \lambda_{j,i} - \sum_{i=0}^{j-1} \lambda_{i,j} = \frac{1}{N+1}, \ \ j = 0, \dots, N-1 \,\right\}. \]
The vertices of this polytope are its spanning trees into \(N\) — one out-arc \(s(j) > j\) per node — so there are exactly \(N!\) of them.
Drori–Taylor sits at the path \(s(j) = j+1\), the averaged subgradient method at the star \(s(j) = N\), and Zamani–Glineur at the barycenter of all \(N!\) vertices.