The minimization problem is $\min\limits_{f_i} \sum^K_{i=1} \|f_i(\mathbf{p})\|_\infty$
Could someone explain how the Fenchel duality is used so the primal-dual formation becomes
$$\min_{f_i} \underset{g_i} \max \sum^K_{i=1} \langle f_i(\mathbf{p}), g_i(\mathbf{p})\rangle $$
$$\text{subject to } \|g_i(\mathbf{p})\|_1 \leq 1$$
Some other details I omitted but might be critical
constraints that appears in both the original problem and the primal-dual formulation:
$$\sum^K_{i=1}f_i (\mathbf{p})=1$$
$f_i(\mathbf{p})\geq 0$
and the definition of $f_i(\mathbf{p})$
$$ f_i(\mathbf{p})= \begin{cases} 1, & \text{if } \mathbf{p}\in\theta_i \\ 0, & \text{otherwise} \end{cases} $$
$\mathbf{p}$ is relaxed to the continuous range $[0, 1]$