a contradiction. Therefore, no solution to exists.

Suppose conversely that $G$ has no disconnected components: for any disjoint union $V = V_1 \cup V_2$ with $V_1, V_2 \neq \emptyset$ there exists an $e \in E$ such that $e \cap V_1 \neq \emptyset$ and $e \cap V_2 \neq \emptyset$. So in particular for every $v, w \in V$ there exists a path of hyperedges $e_1, \ldots, e_n \in E$ such that $v \in e_1$, $w \in e_n$, and $e_{j - 1} \cap e_j \neq \emptyset$ for all $1 < j \leq n$. We can see this by writing $V$ as the disjoint union $\{v\} \cup (V \setminus \{v\})$ which gives us $e_1$ and continuing by induction to obtain $e_{j + 1}$ from the disjoint union $V = (e_1 \cup \ldots \cup e_j) \cup (V \setminus (e_1 \cup \ldots \cup e_j))$ until we reach $w$ (which will happen eventually as each new hyperedge adds at least one new vertex and our hypergraph is finite).

Suppose that for a given $R > 0$, $(x_1, \ldots, x_k) \in U$ satisfies $$\label{eq:energymaxradius}
\max_{1 \leq i < j \leq k} \|x_i - x_j\| \geq R.$$ We will now focus on two vertices for which the relative 