distance in is maximal. As described above there exists a path of hyperedges between these two vertices. So there exist vertices $i_1, \ldots, i_{n + 1} \in V$ and edges $e_1, \ldots, e_n \in E$ such that $\|x_{i_1} - x_{i_{n + 1}}\|$ is maximal, $i_1 \in e_1$, $i_{n + 1} \in e_n$, and $i_j \in e_{j - 1} \cap e_j$ for $1 < j \leq n$. Along this path, we have $$\begin{aligned}
R & \leq \|x_{i_1} - x_{i_{n + 1}}\|
 = \left\|x_{i_1} - z_1 + \sum_{m = 2}^n (z_{j - 1} - x_{i_m} + x_{i_m} - z_j) + z_n - x_{i_{n + 1}} \right\| \\
& \leq \|x_{i_1} - z_1\| + \sum_{m = 2}^n (\|x_{i_m} - z_{j - 1}\| + \|x_{i_m} - z_j\|) + \|x_{i_{n + 1}} - z_n\|.\end{aligned}$$ Hence, one of these $2 \, n$ terms must be at least $R / (2 \, n)$. So there exist a vertex $p \in V$ and a hyperedge $e_q \in E$ such that $p \in e_q$ and $\|x_p - z_q\| \geq R/(2 \, n) \geq R/(2 \, l).$ Therefore, $f$ satisfies $$\label{eq:energyblowupmax}
f(x_1, \ldots, x_k)
\geq \frac{\alpha}{2} \, c_q \, \|x_p - z_q\|^{\gamma}
\geq \frac{\alpha}{2 \, (2 \, l)^{\gamma}} \, \left( 