\in U$ has minimal energy, then necessarily $(0, x_2, \ldots, x_k) \in C$ where $$C := \bigcap_{1 \leq i < j \leq k} \left\{ (x_1 = 0, x_2, \ldots, x_k) \in {\mathbf{R}}^{d \times k} {\ |\ }r \leq \|x_i - x_j\| \leq R \right\}.$$ As $C \subseteq {\mathbf{R}}^{d \times k}$ is both closed and bounded (because $x_1 = 0$), $C$ is compact by Theorem 1.8.17 in [@Duistermaat2004i]. We also have that $C \subseteq U$ and $f$ is continuous on $U$, so $f$ is continuous on the compact set $C$ and therefore $f \vert_C$ attains its minimum at a certain point in $C$ by Theorem 1.8.8 of [@Duistermaat2004i]. Since the minimum of $f$ necessarily lays within $C$ by construction, this is a solution to .

Therefore, in solving , we should restrict ourselves to connected hypergraphs. If the provided hypergraph is not connected, we have to treat each connected component separately, where a solution is guaranteed to exist by .

Algorithm
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$p_v \gets v$; $p_v \gets p_{p_v}$; $p \gets \min \{p_v {\ |\ }v \in e\}$; $p_{p_v} \gets p$; $p_v \gets p_{p_v}$;

We use the above observations to create a multilevel algorithm which approximates a minimum-energy solution. Firstly, we find 